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CONCRETE-STEEL 
CONSTRUCTION 

PART  I— BUILDINGS 


A  TREATISE  UPON  THE  ELEMENTARY  PRINCIPLES  OF  DESIGN 
AND  EXECUTION  OF  REINFORCED  CONCRETE 
WORK  IN  BUILDINGS 


BY 


HENRY  T.  EDDY,  C.  E.,  Ph.  D.,  Sc.  D. 

MEM.  AM.  MATHEMATICAL  SOC. 
MEM.  AM.  PHILOSOPHICAL  SOC. 
MEM.  AM.  PHYSICAL  SOC. 

PROFESSOR  OF  MATHEMATICS  AND  MECHANICS,  COLLEGE  OF 

ENGINEERING,  AND  DEAN  OF  THE  GRADUATE 

SCHOOL,  EMERITUS,  UNIVERSITY  OF 

MINNESOTA 

AND 

C.  A.  P.  TURNER,  C.  E. 

MEM.  AM.  SOC.  C.  E. 
MEM.  CAN.  SOC.  C.  E. 

CONSULTING  ENGINEER,  VANCOUVER,  WINNIPEG, 
MINNEAPOLIS,  NEW  YORK,  CHICAGO,  ETC. 


MINNEAPOLIS 
1914 


^ 


COPYRIGHT  1914 
BY 

H.  T.  EDDY  and  £.  A.  P.  TURNER 

ALL'RIGHTS  RESERVED 


VII 


PREFACE 

When  we  consider  the  fact  that  fire  losses  in  Canada  and  the 
United  States  amount  each  year  to  half  a  billion  dollars,  and  that  the 
question  of  commercial  economy  and  cost  determines  whether  buildings 
shall  be  built  of  fireproof  and  incombustible  materials  such  as  re- 
inforced concrete,  or  of  inflammable  materials  such  as  are  used  in 
timber  construction,  it  is  evident  how  important  it  is  to  the  general 
public  to  be  able  to  determine  on  theoretically  correct  principles 
whether  safe  fireproof  buildings  can  be  built  at  practically  no  greater 
cost  or  even  less  cost  than  combustible  ones.  In  case  of  any  un- 
certainty as  to  theoretical  principles  to  be  applied,  the  designer 
is  compelled  for  safety  to  employ  materials  in  such  lavish  amounts 
as  to  render  cost  prohibitive. 

Engineering  writers  have  heretofore  failed  to  apply  the  mathe- 
matical theory  of  elasticity  to  those  forms  of  reinforced  concrete 
that  differ  from  beams  in  their  manner  of  reinforcement  and  that 
depend  for  their  mechanical  action  on  bond  shear  as  it  is  involved 
in  multiple  way  systems.  The  present  treatise  is  devoted  to  dis- 
cussion of  the  elementary  principles  and  the  practical  problems 
presented  by  building  work  in  reinforced  concrete.  In  it  an 
effort  is  made  to  dissipate  differences  of  opinion  due  to  lack  of 
familiarity  with  the  mechanics  of  reinforced  concrete  such  as  tend, 
at  the  present  time,  to  retard  its  introduction  and  to  hinder  to  some 
extent  the  rapidity  of  its  progress  in  the  commercial  field. 

The  endeavor  is  made  in  this  treatise,  to  bring  out  these  mechan- 
ical laws,  and  to  treat  at  length  the  restraint  imposed  upon  the 
elements  of  the  composite  structure — the  steel  and  the  concrete — 
in  accordance  with  the  fixed  principles  of  physics  and  mechanics 
in  order  that  rational  rules  may  be  more  generally  adopted  for  the 
safe  and  economic  design  of  this  type  of  fireproof  buildings.  Failure 
on  the  part  of  the  engineering  profession  to  consider  these  laws 
in  drawing  up  building  codes,  has  led  to  grave  errors  thereby  offer- 
ing a  premium  on  the  more  dangerous  types  of  designs  in  concrete 
building  work,  and  placing  at  a  disadvantage  the  more  scientific, 
safe,  and  conservative  types  of  work  as  determined  by  the  records 
of  these  types  in  practical  construction. 


3±3Q22 


VIII  PREFACE 

While  the  mechanical  laws  above  referred  to  are  simple  in  the 
extreme  when  once  understood,  their  application  is  so  far  from 
obvious  in  a  superficial  consideration  of  the  subject  that  it  has 
required  continuous  investigation  and  patient  study  on  the  part  of 
the  authors  for  years  of  time  to  determine  and  classify  their  operation 
according  to  the  mechanical  principles  and  laws  of  physics,  and  this 
supplemented  by  expert  observers  engaged  for  many  months  in 
the  conduct  of  experiments  and  tests  to  decide  the  questions  which 
this  continued  study  and  consideration  had  raised. 

In  this  treatise  patented  as  well  as  unpatented  types  have  been 
included  for  the  reason  that  while  the  consulting  engineer  without 
license  has  no  right  to  design  patented  types,  he  is  called  upon  to 
report  upon  their  strength  and  should  for  that  reason  be  as  familiar 
with  their  analysis  as  with  unpatented  constructions. 

The  hope  of  the  authors,  who  have  devoted  so  much  time  and 
expense  to  the  investigation  and  presentation  of  the  fundamental 
principles  explained  in  these  pages,  is  that  as  these  principles  be- 
come more  widely  known  and  understood  needless  accidents  and  loss 
of  life  in  the  erection  of  concrete  building  work  will  be  avoided  and 
unsatisfactory  designs  caused  by  the  failure  of  the  engineering 
profession  at  large  to  understand  and  introduce  into  practice  the 
proper  limitations  of  steel  ratios  as  depending  on  the  relative  thick- 
ness or  depth  of  beam  and  slab  to  span  and  the  correct  arrangement 
and  disposition  of  the  steel  in  the  slab,  will  disappear  from  the 
engineering  field. 

The  frank  avowal  of  this  aim,  carrying  with  it  as  it  does  a  criti- 
cism of  no  small  amount  of  work  by  the  profession  at  large,  might 
be  considered  egotistical  were  the  engineer  authors  unqualified  by 
long  experience  to  speak  with  some  authority,  the  one  as  pro- 
fessionally occupied  with  applied  mathematics,  having  investigated 
and  taught  higher  mathematics  and  structural  mechanics  for  nearly 
fifty  years,  the  other  as  a  bridge  and  structural  engineer  engaged  in 
the  active  practice  of  his  profession  for  twenty-five  years  and  res- 
ponsible for  the  execution  of  two  thousand  concrete  structures  which 
have  been  erected  without  any  serious  accident  to  the  workmen  on 
them  which  could  be  charged  to  the  risk  of  such  erection,  and  this 
notwithstanding  the  fact  that  much  of  this  work  has  been  carried 
on  in  the  unfavorable  temperature  conditions  of  the  northern 
winter  weather. 


PREFACE  IX 

The  requirements  of  economy  in  the  arrangement  and  dissemina- 
tion of  the  reinforcement  have  been  dealt  with  at  considerable 
length.  There  are  many  who  have  the  idea  that  if  they  get  the  steel 
into  the  concrete  somehow  that  is  about  all  that  is  required,  while 
as  a  matter  of  fact,  upon  the  position  and  arrangement  of  the  re- 
inforcement in  the  concrete,  the  strength  of  a  slab  may  readily  vary 
a  hundred  percent  or  more  depending  upon  its  lateral  distribution, 
and  the  stiffness  may  vary  four  hundred  percent,  while  with  the 
vertical  distribution  the  strength  may  vary  fifty  to  eighty  percent 
and  the  stiffness  five  hundred  percent,  and  by  combining  the  vertical 
and  lateral  arrangement  of  the  same  metal  the  strength  may  vary 
four  to  five  hundred  percent,  and  the  stiffness  three  thousand  percent. 

That  these  differences  are  not  understood  by  the  layman  is 
not  surprising,  but  they  should  be  understood  by  the  professional 
engineer  who  has  had  every  opportunity  to  observe  the  deport- 
ment of  finished  structures  under  load,  as  have  those  in  charge  of 
building  departments  thruout  the  country. 

But  aside  from  these  questions  of  relative  arrangement  and 
disposition  of  materials  is  one  underlying  advantage  possessed  by 
concrete  construction  whose  dominating  effect  is  apt  to  be  over- 
looked. That  advantage  inheres  in  the  monolithic  character  of 
this  form  of  construction,  which  imparts  to  it  a  stability  and  strength 
which  has  too  frequently  not  been  properly  taken  into  the  account 
either  by  the  layman  or  the  responsible  designer.  Following  the 
ideas  current  in  ordinary  structural  design  where  the  whole  is  built 
up  by  assembling  and  joining  together  a  number  of  independent 
elements  or  units  these  preconceived  ideas  have  led  to  the  attempt 
to  analyse  these  monolithic  structures  into  separate  members  which 
are  assumed  to  act  independently  as  they  do  in  steel  structures. 
Such  assumptions  lead  to  conclusions  that  have  little  relation  to 
the  facts  as  shown  by  tests  and  by  experience  as  well. 

The  Authors. 


CORRECTIONS 


CONCRETE-STEEL  CONSTRUCTION 

BY 
EDDY  and  TURNER 


Page     1,  line  23,  write  maximum  instead  of  maxium. 
"       4,    "     17,  write  make  instead  of  made. 
"     41,    "       3,  write  ledger  to  ledger  instead  of  ledger  ledger. 
"     41,    "     36,  write  length  instead  of  width. 
"     58,    "     14,  write  corresponding  for  corrected. 


75,  Equation  (4)  write  instead  of    —  ^s 

k  j  k 

82,  "  12,  write     ABl     instead  of      AB. 

82,  "  20,  write      T      instead  of     TI. 

82,  "  20,  write     Es     instead  of     Es. 

82,  "  22,  write      T      instead  of     TV 

82,  "  23,  write     OB1      instead  of      B^ 

83,  "  31,  write  moderate  instead  of  moderate. 

87,  last  line  on  page,  write   n  =  15  instead  of  p  =  15. 

100,  12th  line  from  bottom  of  page  write  tabular  instead 

of  tabular. 
106,  Title  of  Fig.  42,  write  Test  in  which  Arch  Action 

instead  of  Test  in  which  Action. 
123,  Fig.  46,  write    L     instead  of    7. 

135,    "     13,  write  seventy  instead  of  esventy. 
183,    "     25,  write  Turneaure  instead  of  Turnearue. 
280,    "       9,  write  along  y  instead  of  along  n. 

296,  "       7,  write  5/8  instead  of  7/8. 

297,  "       6,  write  2  (a2  +  62)     instead  of     (2a2  +  b2) 

318,    "     21,  write    guarantee   of    working    strength    instead    of 
guarantee  of  strength  for  working. 

427,  "     13,  write  Macomber  instead  of  Curtiss. 

428,  "       1,  write  mechanism  instead  of  mechansim. 


XI 


CONTENTS 

CHAPTER  I. 

PAGE 

1.  Introductory  2.  Historical.  3.  Materials;  Portland 
Cement  Specifications.  4.  Quick  Tests;  Accelerated  Test; 
Care  of  Cement.  5.  Specification  for  Aggregate;  Sand;  Test- 
ing Sand;  Gravel;  Stone.  6.  Proportions  of  Materials. 
7.  Analysis  of  Strength  of  Concrete.  8.  Hardening  of  Port- 
land Cement;  Increase  of  Strength  with  Age;  Coefficient  of 
Expansion.  9.  Bond  between  Concrete  and  Steel.  10.  Var- 
iation in  Strength  of  Concrete  with  Variation  of  Tempera- 
ture and  Moisture.  11.  Machine  Mixing.  12.  Pouring  Con- 
crete. 13.  How  to  Determine  when  Concrete  is  thoroly 
Cured.  14.  Handling  Concrete  above  Freezing;  Concrete 
below  Freezing.  15.  Action  of  Salt.  16.  Curing  Concrete 
where  Proper  Precautions  have  not  been  Taken.  17.  Pre- 
cautions in  Splicing,  Mixing,  Heating,  etc.  18.  Caution 
Regarding  Removal  of  Forms.  19.  Reinforcing  Steel;  Speci- 
fications. 20.  Quality  of  Steel.  21.  Cold  Bending  with  Mild 
Steel.  22.  Bending  Machines.  23.  Hot  Bending  and  Pre- 
cautions with  High  Carbon  Steel.  24.  Centering;  Partial 
Removal  of  Forms 1-48 

CHAPTER  II 

1.  Classification  of  General  Types  of  Floor  Construction. 
2.  Utility  of  the  Theory  of  Action  of  Structures.  3.  Prin- 
ciple of  Proportion.  4.  Variation  in  Strength  with  Thick- 
ness. 5.  Variation  in  Strength  with  Span.  6.  Theoretical 
Treatment  by  Proportion  of  Mushroom  Slabs,  Beams  and 
Slabs  Reinforced  Two  Ways  Supported  on  Four  Sides.  7.  Ex- 
ample of  Computation  for  Deflection  of  Mushroom  Floor 
Slabs.  8.  Rectangular  Slab  Supported  by  Girders  on  Four 
Sides.  9.  Computation  of  Deflection  Applied  to  Practical 
Examples;  Slab  Supported  on  Four  Sides  by  Girders.  10. 
Short  Span  Slab  and  Arch  Action  that  may  be  Counted  up- 
on in  Their  Use.  11.  Value  of  Finish  Coat,  Strip  Fill  and 
Wood  Floor  from  the  Standpoint  of  Deflection 49-  67 


XII  CONTENTS 

CHAPTER  III 

BEAMS 

1.  Elastic  Properties  of  Materials,  Concrete  and  Steel.  PAGE 
2.  Tensile  Strength.  3.  Elasticity  of  Concrete.  4.  Concrete 
Beams.  5.  Modulus  of  Elasticity.  6.  Formulas  for  Rein- 
forced Concrete  Construction.  7.  Determining  Moment. 
8.  Discussion  of  the  Elastic  Properties  of  Beams  and  Assum- 
tions  involved  in  the  Preceding  Theory.  9.  Classification  of 
Beams.  10.  Economic  Design  of  Beams.  11.  Safe  Loads 
for  and  Tests  of  Reinforced  Concrete  Construction.  12.  True 
and  Nominal  Factor  of  Safety.  13.  Method  of  Loading  for 
Tests.  14.  Shears  in  Beams.  15.  Shear  and  Diagonal  Ten- 
sion. 16.  Working  Stresses.  17.  Compound  Tensile  Strength. 

18.  The    Reinforced    Concrete    Beam    as    a    Mechanism. 

19.  Bond  Shear  in  Blocks.    20.  Bond  Shear  in  Splices.    21. 
Bond  Shear  in  Beams.    22.  Bond  Shear  in  Slabs.    23.  Mech- 
chanics  of  Embedment.    24.  Bond  with  Deformed  Bars ....   68-133 

CHAPTER  IV 

BEAM    ACTION    AND    SLAB    ACTION    COMPARED  THROUGH    APPLICATION 
OF    THE    LAWS    OF    BOND    SHEAR    AND    THE    THEORY    OF    WORK. 

1.  Introductory.  2.  Slab  as  a  Mechanism.  3.  Relative 
Stiffness  of  Beam  and  Slab.  4.  Indirect  Tension.  5.  Com- 
parison of  Deflections  at  Mid  Span  of  the  Diagonal  Belt  and 
the  Direct  Belt.  6.  Bending  Moments.  7.  Continuity  in  Flat 
Slabs  and  in  Thin  Slab  on  Beams  Contrasted  Experimentally. 
8.  Variation  in  the  Position  of  the  Line  of  Inflection  in  Con- 
tinuous Flat  Slab  Construction.  9.  Effect  of  Adding  Finish 
to  the  Rough  Slab.  10.  Illustrative  Example.  11.  De- 
pressed Head  or  Drop 134-157 

CHAPTER  V 

THEORY    OF    FLAT    SLABS 

1.  Flat  Slab  Floors.  2.  Notation.  3.  True  and  Apparent 
Bending  Moments.  4.  Poisson's  Ratio.  5.  General  Differ- 
ential Equation  of  Moments.  6.  General  Differential  Equa- 
tion of  Deflections.  7.  Solution  of  the  Differential  Equation 
in  Case  of  Uniform  Slab  Supported  on  Columns.  8.  Solution 
for  Side  Belts.  9.  Practical  Formulas  for  Stresses  in  Side 
Belts.  10.  Practical  Formulas  for  Stresses  in  Column  Heads. 
11.  Practical  Formulas  for  Stresses  in  the  Middle  Area  of 
Panel.  12.  Deflections  at  Mid  Span  of  the  Side  and  Diagonal 


CONTENTS  XIII 

CHAPTER  V— Continued 

Belts.    13.  Proportionate  Deflections  at  Mid  Span  and  Cen-     PAGE 
ter  of  Panel.    14.  Radial  and  Ring  Rods,  and  Shear  around 
Cap.    15.  Standard  Mushroom  System  and  Other  Systems  158-219 

CHAPTER  VI 

COMPUTED    STRESSES    AND    DEFLECTIONS    VERIFIED    BY    TESTS 

1.  Specimen  Computations  of  Stresses  and  Deflections 
in  Several  Slabs.  2.  Further  Calculations  of  Test  Slabs. 

3.  Comparative    Test    of   Norcross    and   Mushroom   Slabs. 

4.  Investigation  of  Structures  by  the  Berry  Extensometer 

and  Interpretation  of  Results 220-268 

CHAPTER  VII 

MOMENTS   IN  TWO-WAY  AND  FOUR-WAY  FLAT  SLABS 

1.  Simple  Approximate  Theory  of  Four-Way  Slabs. 
2.  Simple  Approximate  Theory  of  Two-Way  Slabs.  3. 
Weight  of  Steel  in  Two-Way  and  Four-Way  Slabs  Compared. 

4.  Panels   Reinforced  unequally  Lengthwise  and  Crosswise. 

5.  Slab  with  Rectangular  Panels  Supported  on  Beams  or 
Walls.    6.  Steel  Ratios  and  Minimum     Thickness  of  Slabs. 

7.  Size   and  Spacing   of   Rods   in   Flat   Slab   Construction. 

8.  Hollow    Tile    and    Concrete    Construction    for    Floors. 

9.  Slabs  Reinforced  with  Expanded  Metal 269-302 

CHAPTER  VIII 

REINFORCED    CONCRETE    COLUMNS 

1.  General  Considerations  Governing  Distribution  of  Re- 
inforcement and  Placing  of  Material;  Types  of  Columns. 

2.  Consideration  of  Safety  in  Determining  Carrying  Loads. 

3.  Experimental  Data;  Tests  at  Phoenixville ;  Tests  by  Bach. 

4.  Reinforced  Columns  Classified  by  the  Manner  in  Which 
the    Loads    are    Applied.    5.  Considere    Formula.    6.  Safe 
Ultimate  Limit  of  Compression.    7.  Mode  of  Operation  of 
Reinforcement    in    Concrete  Columns.      8.  Effect  of  Hoop- 
ing.   9.  Comparison  of  Test  Data.    10.  Formula  for  Columns 
where  Load  is  Brought  upon  the  Steel  by  Direct  Bearing  on 
the  Metal.    11.  Working  Stresses.    12.  Structural   Columns 
Filled  with  Concrete.    13.  Concrete  Columns  Compared  with 
Structural  Steel.    14.  Wall  Columns  and  Interior  Columns 
in    Skeleton    Construction.      15.     Temperature    Effect     on 
Columns.    16.  Economic  Column  Design 303-330 


XIV  CONTENTS 

CHAPTER  IX 

FOUNDATIONS 

1.  Bearing  Value  on  the  Soil.    2.  Column  Footings  and     PAGE 
Method  of  Figuring.    3.  Pile  Foundations.    4.  Driven  Piles. 
5.  Piles  Cast  in  Place.    6.  Safe  Bearing  Loads  for  Piles 331-342 

CHAPTER  X 

ELEMENTS    OF    ECONOMIC    CONSTRUCTION    AND    COST    OF    REINFORCED 

CONCRETE    WORK 

1.  Introductory.  2.  Column  Spacing.  3.  Floors,  Strength 
and  Stiffness.  4.  Centering.  5.  Columns.  6.  Bearing  Walls 
or  Full  Concrete  Skeleton.  7.  Concrete,  or  Brick  Exterior 
Walls.  8.  Rich  Mixture.  9.  Economy  in  Selecting  Aggre- 
gate. 10.  Cinders.  1 1 .  Adaptability .  12.  Rapidity  of  Erection 
and  Ease  of  Securing  Materials.  13.  Analysis  of  Items  of 
Cost.  14.  Labor,  Unit  Prices,  Quantities  of  Material.  15. 
Cost  of  Steel.  16.  Cost  of  Bending.  17.  Cost  of  Hooping 
for  Columns.  18.  Cost  of  Centering;  Cost  of  Framing. 
19.  Season  of  Year.  20.  Dead  Charges.  21.  General  Data 
on  Cost.  ..343-363 


CHAPTER  XI 

1.  Fireproof  Properties  of  Concrete  and  the  Protection  of 
Steel  from  Heat.  2.  Fire  Tests.  3.  The  Theory  of  Fire  Pro- 
tection. 4.  Terra  Cotta  and  Tile  Compared  with  Concrete. 
5.  Rates  of  Insurance  on  Concrete  Buildings  and  Contents .  364-374 


CHAPTER  XII 

1.  Protection  of  Steel  and  Iron  from  Corrosion  by  Port- 
land Cement.  2.  Permanence  of  Concrete  Construction 
when  made  with  Proper  Materials.  3.  Concrete  Mixed 
Dry  and  Tamped.  4.  Hair  Cracks,  Map  Checks,  and  Craz- 
ing. 5.  Temperature  Effects.  6.  Disintegration  of  Concrete 
by  Oil,  Grease,  etc.  7.  Electrolysis 375-387 


CONTENTS  XV 

CHAPTER  XIII 

1.  Floor  Finish.    2.  Strips  and  Strip  Fill  for  Wood  Floors.      PAGE 

3.  Width  of  Flooring.    4.  Cement  Finish  Coat.    5.  Mixture 
of  Finish.    6.  Hardening  Compound.  7.  Treatment  of  Floors. 
8.  Concrete  Stairs.    9.  Insulation  of  Roofs.    10.  Protection 
and  Provision  for  Plumbing.    11.  Placing  Electric  Conduits, 
Gas    Pipe,    etc.    12.  Plastering    on    Reinforced    Concrete. 

13.  Suspended  Ceilings 388-396 

CHAPTER  XIV 

1.  Artistic  and  Commercially  Practicable  Concrete  Sur- 
face Finishes.  Stipple  Coat.  2.  Plaster  Coat  on  Rough  Cast 
Concrete.  3.  Finish  Obtained  by  Brushing  and  Washing. 

4.  Finish  by  Tooling.    5.  Cast  Stone 397-408 

CHAPTER  XV 

1.  The  Execution  of  Work;  Hardening  of  Concrete; 
Pouring  Concrete;  Separation  of  Materials;  Test  for  Hard- 
ness in  Warm  Weather;  Test  for  Hardness  in  Cold  Weather; 
Lap  of  Reinforcement  over  Supports.  2.  Responsibility  of 
the  Engineer.  3.  Responsibility  of  the  Constructor  and 
Engineer  Superintendent.  4.  Significance  of  Cracks  in  Re- 
inforced Concrete.  5.  Encouragement  to  Progress  in  the 
Concrete  Industry  by  Patents;  Scope  of  Patents;  Importance 
of  Investigating  the  Scope  of  a  Patent;  Prior  Art;  Genus  and 
Species  Patent;  Consequences  of  Infringement 409-431 


CONCRETE  STEEL  CONSTRUCTION 

CHAPTER  1. 
I.     Introductory 

The  history  of  structural  engineering  as  a  science  dates  from  the 
early  part  only  of  the  last  century.  The  progress  made  has  been 
remarkable  indeed,  and  the  materials  mainly  used  have  varied  dur- 
ing well-defined  periods.  Up  to  1860,  timber,  wrought  and  cast 
iron  were  mainly  used;  from  1860  on  wrought  iron  with  some  cast  iron 
was  generally  employed  in  bridges  and  other  engineering  structures; 
from  1890  to  the  present  time  steel  has  replaced  wrought  iron;  and 
while,  for  long-span  bridges,  it  will  perhaps  be  some  time  before  a 
more  suitable  metal  is  found,  yet  for  short  spans,  buildings,  ware- 
houses and  the  like,  the  enterprise  of  the  manufacturers  of  Portland 
cement  has  placed  at  the  disposal  of  the  engineer  a  new  material, 
reliable,  if  properly  handled,  and  of  reasonable  cost,  which  bids 
fair  to  largely  supplant  steel  in  the  construction  of  minor  engineering 
works.  Indeed  today,  a  warehouse  designed  for  a  capacity  of  400 
pounds  per  square  foot  of  floor,  columns  16  to  24  feet  centers,  can 
be  built  more  cheaply  of  reinforced  concrete  than  of  wood  frame  and 
floor,  with  similar  brick  walls.  Where  the  strength  required  is  less, 
timber  at  the  present  rate  is  slightly  cheaper,  since  the  cost  of  center- 
ing is  the  same  for  light  as  for  heavy  construction.  Still,  the  differ- 
ence is  so  slight  that,  considering  the  saving  in  insurance,  owners 
will  shortly  be  convinced  that  they  cannot  afford  to  continue  the  con- 
struction of  fire  traps  if  they  are  to  realize  the  maxium  profit  on 
their  investment. 

The  strength  of  Portland  concrete  in  compression  is  equal  to 
that  of  good  building  stone,  with  the  advantage  that  it  can  be  placed 
in  monolithic  masses.  Its  tensile  strength,  like  stone,  is  greatly 
inferior  to  that  in  compression.  Concrete  yields  but  little,  the 
stretch  being  confined  to  the  weak  section.  When,  however, 
steel  is  embedded  in  the  concrete  and  properly  disseminated  thro 
it,  the  deformation  or  stretch  is  distributed  greatly  by  the  metal. 

The  conditions  leading  to  the  combination  of  concrete  and  steel 
in  a  beam  or  girder  are  these:  Concrete  is  an  excellent  and  trust- 
worthy material  for  compression  and  steel  for  tension.  Hence 


2  HISTORICAL 

steel  should  be  distributed  in  such  manner  as  to  carry  the  tensile 
stresses  of  the  chord  and  web.  To  do  this  economically  we 
can  reason  from  analogy  of  a  truss  or  beam.  The  further  from  the 
neutral  axis  the  more  effective  the  steel  section,  hence  the  reinforce- 
ment for  tensile  chord  stress  should  be  at  the  bottom  of  the  beam  or 
as  close  to  it  as  satisfactory  protection  against  heat  by  fire  will 
permit.  Now  the  beams  in  a  building  are  of  constant  section,  and 
since  a  continuous  beam  is  stiffer  and  stronger  than  a  beam  of  the 
same  section  discontinuous  over  supports,  the  ideal  concrete-steel 
beam  should  be  continuous  and  the  top  flange  reinforced  over 
supports. 

As  Morsch  states  in  his  treatise: — 

"  Practice  has  been  far  ahead  of  theory.  The  principal  question 
in  controversy  has  been  whether  the  tensile  strength  of  concrete  in 
bending  should  be  considered.  Among  practical  builders  this  was 
decided  at  the  start,  and  -decided  against  its  inclusion,  because 
absolutely  no  attention  is  paid  to  it  and  the  steel  is  stressed  to  the 
safe  limit.  The  tensile  strength  of  the  concrete  is  entirely  ignored. 
On  this  assumptiDn  is  based  the  first  method  of  the  theoretical 
computation  of  slabs  devised  by  Koenen  in  Berlin  in  1866  and  his 
method  has  been  used  by  the  majority  ever  since." 

2.      Historical 

Steel  and  concrete  as  a  combination  of  materials  for  engineering 
structures  is  much  older  than  is  generally  supposed.  Professor  Barbour 
of  the  University  of  Nebraska,  in  a  very  interesting  lecture  delivered 
to  the  Cement  Users  of  the  State  of  Nebraska,  described  certain 
flat  arches  discovered  in  the  ruins  of  ancient  Rome,  which  were  for 
a  long  time  a  puzzle  to  engineers  and  architects  until  it  was  found 
that  trrey  were  tied  together  and  the  thrust  in  large  part  resisted 
by  iron  tie  rods.  So  far  as  known,  little  was  done,  however,  in  the 
way  of  combining  the  old  Roman  concrete  with  iron,  except  in  the 
isolated  instance  just  cited,  and  it  was  not  until  1855  that  iron  was 
combined  with  concrete  in  a  manner  similar  to  that  in  which  it  is 
utilized  at  the  present  time. 

At  the  Paris  Exposition  of  1855,  Lambot  exhibited  a  boat  made 
of  reinforced  concrete,  while  Francois  Coignet  is  credited  with 
having  built  floors  and  pipes,  in  the  construction  of  which  he  had 
combined  steel  and  concrete  to  some  extent. 

In  1867,  Scott,  a  Lieutenant  Colonel  of  Engineers  of  the  British 
Army,  took  out  a  British  patent  on  concrete  floor  slabs  reinforced 
in  one  direction,  and  also  in  two  directions,  and  in  some  of  the 
drawings  in  this  early  patent  woven  fabric  is  found  combined  with 
rods. 


WORK   OF    MONIER   AND    SCOTT  3 

In  France,  Joseph  Monier  took  out  patents  about  the  same 
time,  and  to  him,  perhaps  more  than  to  any  other  person,  is  to  be 
given  credit  for  the  commercial  introduction  of  reinforced  concrete 
on  a  large  scale.  His  first  use  of  this  type  of  construction  was  to 
fabricate  large  plant  tubs  which  he  found  more  durable  than  those 
of  wood,  and  more  readily  transported  than  those  of  cement  without 
reinforcement.  In  1867  he  took  out  his  first  French  patent,  which 
he  soon  followed  with  a  number  of  others  on  reservoirs,  floors,  and 
straight  and  arched  beams  in  combination  with  the  floors,  etc.  In 
1884  the  Monier  patents  were  purchased  by  the  firms  of  Freitag 
and  Heidschuch  in  Newstad-on-the-Haardt  and  Martinstein  and 
Josseaux  in  Offenbach-on-the-Main.  Later  the  patent  rights  in 
Germany  were  sold  to  Engineer  Wayss,  under  whose  supervision 
tests  were  made  in  Berlin,  the  results  of  which  were  published  in 
1887,  and  on  the  basis  of  these  experiments  Wayss  succeeded  in  intro- 
ducing the  Monier  system  into  many  structures. 

The  Scott  patent  is  especially  significant  since  the  specification 
states  that  tie  rods  and  hoop  iron  were  to  take  the  tensile  strains  and 
concrete  the  compressive  stresses,  showing  clearly  that  Scott  under- 
stood the  basic  principle  of  reinforced  concrete.  Little  was  done, 
however,  by  the  earlier  inventors  in  developing  a  working  theory 
of  design. 

Herr  Wayss  conducted  certain  tests  in  Berlin  and  published  his 
results  in  a  pamphlet  entitled  "Das  System  Monier,  Eisengerippe 
mit  Zementumhullung."  In  this  pamphlet  Wayss  expressed  the 
opinion  that  the  steel  must  be  placed  where  the  tensile  stresses  oc- 
curred. The  tests  were  witnessed  by  government  officials  as  well 
as  by  private  engineers  and  architects.  Government  Architect 
Koenen,  now  Director  of  the  Actiengesellschaft  fur  Beton-und 
Monierbauten,  in  Berlin,  was  commissioned  by  Wyass  to  work  out 
methods  of  computation  from  these  tests,  which  were  published  in 
the  volume  of  the  "Zentralblatter  der  Bauverwaltung"  for  1886 

Morsch,  Concrete  Steel  Construction,  1907  thus  comments  on 
the  introduction  of  reinforced  concrete  at  this  period  (1886). 

"Commencing  at  that  time  (1886)  a  theoretical  foundation 
was  evolved,  according  to  which  the  design  of  reinforced  concrete 
work  would  be  effected,  and  through  these  preliminary  labors, 
this  method  of  construction  was  extensively  adopted  in  Germany 
and  Austria.  A  turning  point  in  its  development  was  the  Interna- 
tional Exposition  in  Paris,  in  1900,  and  the  report  by  von 
Emperger,  published  at  that  time  in  regard  to  the  position  which 
the  subject  occupied. 

Because  of  the  scientific  investigation  of  reinforced  concrete 
during  the  past  few  years,  it  has  made  rapid  progress  in  Germany. 


4  WORK  OF  HYATT,  RANSOME,  ET  AL 

It  was  specially  promoted  by  the  publication,  in  1904,  through 
the  cooperation  of  experts  and  practical  men  of  the  "Leitsatze" 
of  the  Verbands  Deutscher  Architekten-  und  Ingenieurvereine  and 
the  Deutschen  Betonvereins  as  well  as  by  the  Regulations  of  the 
Prussian  Government,  which  abolished  many  restrictive  rules, 
cleared  the  way,  and  inspired  in  the  widest  circles  confidence  in 
the  new  method  of  building." 

In  1870  Phillip  Brannon  made  what  appears  to  have  been  the  first 
application  for  an  English  patent  on  reinforced  concrete  piles.  The 
patent  was  granted  in  1871,  showing  reinforced  concrete  piles  with 
longitudinal  reinforcement  of  angle  irons  united  by  bars  riveted  across 
them,  the  whole  being  wound  spirally  with  wire. 

Thaddeus  Hyatt  between  the  years  1873  and  18S1  took  out 
between  thirty  and  forty  different  patents  relating  to  reinforced 
concrete  work,  pavement  lights,  floors  and  slabs.  It  does  not  appear, 
however,  that  Hyatt  made  a  success  of  concrete  construction  com- 
mercially altho  he  did  made  a  success  of  his  paving  lights.  Hyatt 
regarded  a  reinforced  concrete  beam  as  one  corresponding  to  a  steel 
beam,  and  he  considered  the  rods  as  equivalent  to  the  beam  flange 
and  the  concrete  as  the  top  flange,  assuming  the  neutral  axis  at  mid 
depth  of  the  beam. 

A  most  important  patent  was  granted  to  Hyatt  in  1874,  in  Eng- 
land, No.  1715,  in  which  is  disclosed  spiral  and  vertical  reinforce- 
ment for  columns,  which  strangely  enough  was  not  appreciated 
until  attention  had  been  called  to  this  type  of  column  by  its  rein- 
vention at  a  later  date  by  Considere,  to  whom  the  engineering 
profession  is  indebted  for  the  attention  which  he  directed  to  it  by 
the  valuable  tests  carried  out  by  him. 

During  this  time  there  was  considerable  activity  in  the  United 
States.  E.  L.  Ransome  was  building  reinforced  concrete  warehouses 
as  early  as  1884,  and  patented  in  the  United  States  a  twisted  bar 
reinforcement. 

In  1883  John  F.  Golding  secured  an  American  patent  for  ex- 
panded metal  which  was  employed  as  lathing  for  plaster  in  lighter 
gages,  and  as  reinforcsment  for  concrete  slabs  with  larger  mesh 
and  a  No.  10  or  heavier  gage. 

In  Cassell's  Reinforced  Concrete,  published  in  1913,  the  following 
sketch  is  given  of  the  work  of  the  early  pioneers  of  the  art  before 
1900,  Edmond  Coignet  and  Francois  Hennebique:— 

"The  former  of  whom,  by  applying  the  known  principles  of 
mechanics,  evolved  a  system  of  calculation  that  has  proved  re- 
markably truthful,  and  the  latter  of  whom,  basing  his  methods 
of  calculation  upon  results  obtained  in  practice,  has  also  made 
extremely  important  contributions  to  the  technical  consideration 


COIGNET    AND    HENNEBIQUE  5 

of  the  subject.  Coignet  as  the  scientific  investigator,  and  Hen- 
nebique as  commercial  organizer,  are  properly  regarded  as  'the 
pioneers  of  the  modern  evolution  in  the  art  of  building.'  The 
story  has  often  been  told  of  the  opposition  which  Coignet  had  to  fight 
in  getting  the  masonry  of  the  proposed  new  system  of  main 
drainage  in  Paris  in  1892  replaced  by  reinforced  concrete.  He 
promised  a  large  saving  of  money  and  of  time  required  for  con- 
struction, and  his  system,  which  was  finally  adopted,  was  carried 
out  with  complete  success.  Hennebique,  having  organized  a 
technical  staff  and  licensed  a  large  number  of  the  most  influential 
contractors  to  work  his  system,  was  able  to  secure  between  the 
years  1892  and  1899  work  to  the  total  value  of  two  million  sterling, 
representing  three  thousand  constructions,  among  the  most  re- 
markable of  these  being  the  bridge  of  Chatellerault,  460  ft.  long, 
comprising  three  arches,  two  of  133  ft.  span  and  one  of  167  ft. 

Hennebique's  first  patent  dates  from  1892,  (British  patent,  No. 
14,530),  and  in  this  he  demonstrates  the  utility  of  stirrups  to 
reinforce  beams  against  shear,  in  which  matters  he  had  to  an  extent 
been  anticipated  by  Hyatt  in  1877  and  Meyenberg  in  1891.  In 
1897  Hennebique  introduced  cranked-up  rods,  and  placed  these 
one  above  the  other,  so  as  to  reduce  the  width  of  the  beam,  follow- 
ing (to  some  extent)  the  lines  laid  down  by  Hyatt  in  1877  and 
F.  G.  Edwards  in  1892,  in  which  latter  year  M.  Koenen  and  G.  A. 
Wayss,  of  Germany,  patented  in  England  a  method  of  floor  con- 
struction with  rods  cranked-up  at  the  point  of  contraflexure,  "the 
parts  in  tension  being  strengthened  by  roughened  or  serrated 
metal  rods  or  strips  embedded  in  the  structure." 

We  have  noted  the  early  work  of  Ransome  in  the  United  States. 
In  his  work  parallel  joists  about  3  inches  wide,  spaced  three  feet 
centers  were  frequently  used,  while  narrow  intersecting  ribs  about  ten 
feet  centers  in  two  direct  ions  surmounted  by  a  thin  slab  were  also 
employed. 

From  1890  to  1900  while  cement  was  relatively  high  in  price, 
reduction  of  mass  at  increased  expense  in  form  work  was  to  be  ex- 
pected. 

The  expanded  metal  companies  introduced  a  large  amount  of 
short  span  concrete  floors  on  structural  steel  frame  six  to  ten  feet 
center  to  center  of  beams,  competing  with  hollow  tile  arches  then 
more  commonly  employed  for  fireproof  construction. 

The  Roebling  Company,  manufacturers  of  wire,  were  at  the  same 
time  putting  in  a  large  amount  of  short  span  arches  and  slabs 
reinforced  with  wire  fabric  and  rods. 

From  1900  to  date,  reinforced  concrete  building  construction  has 
increased  with  wonderful  rapidity,  encouraged  by  the  enterprise 
of  the  manufacturers  of  Portland  cement  in  placing  at  the  disposal 
of  the  constructor  a  reliable  and  uniform  product  at  so  low  a  cost 
that  a  most  powerful  impetus  was  thus  furnished  for  the  more 
complete  development  of  the  commercial  possibilities  of  reinforced 
concrete  in  building  construction. 


O  THEORY,    HISTORICAL 

The  skepticism  of  building  departments  and  the  natural  an- 
tagonism of  tile  interests  forced  the  advocates  of  concrete  construc- 
tion to  make  innumerable  tests  with  the  final  result  of  creating  con- 
fidence in  the  construction  when  properly  designed  and  executed. 

During  the  six  years  subsequent  to  1900 — a  short  period  of  time 
but  an  epoch  from  the  standpoint  of  progress  in  concrete  construc- 
tion— numerous  beam  theories  were  proposed  and  discussed. 

Engineering  opinion  gradually  crystallized  in  the  adoption  of  a 
modification  of  the  common  theory  of  flexure  and  the  assumption 
of  the  linear  law  of  distribution  of  stress  for  purposes  of  computation 
of  beams  and  slabs,  and  this  opinion  has  been  embodied  in  nearly 
all  building  codes  in  cities  throughout  the  United  States  and  Canada. 

In  1904  a  Joint  Committee  of  the  American  Association  of 
Cement  Manufacturers,  American  Society  for  Testing  Materials, 
American  Society  of  Civil  Engineers,  American  Railway  Engineering 
and  Maintenance  of  Way  Association,  was  appointed  to  investigate 
and  report  on  concrete  and  reinforced  concrete.  After  eight  years, 
a  report  was  rendered,  for  which  see  Eng.  News.  Feb.  6,  1913. 

The  report  specifically  states  that  it  does  not  go  into  all  types 
of  construction  or  all  the  applications  to  which  concrete  and  re- 
inforced concrete  may  be  put,  *  *  *  *  *  it  is  not  a  specification 
but  may  be  used  as  a  basis  for  specifications." 

Treatment  of  natural  types  of  reinforced  concrete  is  lacking  in 
this  report.  The  treatment  of  beams,  however,  embodies  the  crystal- 
lized opinion  above  referred  to  and  will  be  referred  to  more  at  length 
later. 

That  a  theoretical  treatmen  t  of  concrete  after  the  manner  of  struc- 
tural iron  work,  although  making  allowance  for  the  elastic  properties 
of  the  two  materials,  is  unsatisfactory  was  early  recognized  by 
practical  men  and  also  by  some  theoretical  writers. 

Marsh  in  his  treatise  on  Reinforced  Concrete,  Edition  of  1905, 
Part  V.  p.  209,  makes  the  following  remarks  on  this  subject: 

"When  properly  combined  with  metal,  concrete  appears  to 
gain  properties  wnich  do  not  exist  in  the  material  when  by  itself, 
and  although  much  has  been  done  by  various  experimenters  in 
recent  years  to  increase  our  knowledge  on  the  subject  of  the  elastic 
behavior  of  reinforced  concrete  we  are  still  very  far  from  having 
a  true  perception  of  the  characteristics  of  the  composite  material. 

"It  may  be  that  we  are  wrong  from  the  commencement  in 
attempting  to  treat  it  after  the  manner  of  structural  iron  work 
and  that  although  the  proper  allowances  for  the  elastic  properties 
of  the  dual  material  is  an  advancement  on  the  empirical  formula 
at  first  employed  and  used  by  many  constructors  at  the  present 


THEORY,    HISTORICAL  7 

time,  yet  we  may  be  entirely  wrong  in  our  method  of  treatment. 
''The  molecular  theory,  i.  e.  the  prevention  of  molecular  defor- 
mation by  supplying  resistances  of  the  reverse  kind  to  the  stresses 
on  small  particles,  may  prove  to  be  the  true  method  of  treatment 
for  a  composite  material  such  as  concrete  and  metal.  This  theory  is 
the  basis  of  the  Cottancjn  construction  which  certainly  produces  good 
results  and  very  light  structures,  and  M.ConsicUre's  latest  researches 
on  the  subject  of  hooped  concrete  are  somewhat  on  these  lines." 

.  In  this  statement  of  Marsh  there  is  some  idea  of  a  possible  new 
principle  of  action;  but  unfortunately  he  was  unable  to  form  and 
express  any  conception  of  how  this  might  operate  in  accordance  with 
mechanical  principles  in  a  manner  which  would  be  of  benefit  to  the 
industry  at  large,  or  aid  in  the  discussion  of  the  stresses  operating  in 
reinforced  concrete. 

The  practical  constructor  has  an  advantage  over  the  theorist  in 
this  respect;  having  observed  the  results  obtained  by  a  new  principle 
he  immediately  profits  by  it  by  taking  advantage  of  the  results 
through  application  of  the  principles  of  simple  proportion  leaving 
the  explanation  as  a  matter  of  academic  interest  to  follow  in  the 
wake  of  his  practical  accomplishment.  In  this  simple  manner  two 
thousand  structures  of  the  Mushroom  type  have  been  erected  and 
tested  for  strength  and  deflection  before  a  comprehensive  scientific 
explanation  of  the  mode  of  operation  was  forthcoming. 

The  constructor  wants  the  least  theory  possible  and  that  the 
simplest  to  meet  the  specific  requirements  of  the  work  he  has  in  hand. 
The  scientist  with  broader  and  more  comprehensive  vision  sees  in 
the  specific  performance  of  the  builder  only  a  special  case  to  be  treated 
in  conjunction  with  the  species  to  which  it  belongs.  That  the  latter 
treatment  is  by  far  the  more  difficult — and  when  correctly  carried 
out  more  valuable  and  satisfactory — compensates  for  the  almost 
inevitable  position  that  the  scientist  occupies  in  a  new  art,  following 
in  the  wake  of  the  practical  constructor,  whose  simple  needs  require 
special  rather  than  general  solutions  of  the  problems  at  hand. 

As  the  authors  now  view  the  art  broad  general  solutions  of  the 
problems  in  reinforced  concrete  are  in  order  and  the  character  and 
nature  of  the  new  properties  added  to  concrete  by  the  dissemination 
of  steel  through  it  brought  out  in  the  following  pages  it  is  hoped  may 
harmonize  many  differences  in  engineering  opinion  existing  at  the 
pressnt  time  relative  to  the  more  advanced  forms  of  construction. 

In  the  discussion  of  concrete-steel  construction,  we  must  consider, 
first,  the  action  of  the  concrete  with  the  steel,  the  function  of 
each  in  the  combination,  the  problems  presented  by  beams,  slabs, 


8  CEMENT    SPECIFICATIONS 

and   columns  separately,  and  finally  the  mix  of  the  concrete  and 
questions  of  cost  in  convenient  placing  of  the  reinforcement. 

Before  taking  up  these  points  in  detail  it  would  seem  in  order, 
however,  to  turn  our  attention  to  the  concrete,  and  the  materials 
entering  into  it,  their  characteristics,  value  and  fitness  and  the 
proper  proportions  to  use. 

3.     Materials 

Portland  Cement  only  should  be  used  in  a  reinforced  concrete 
frame  or  structure.  The  following  specification  is  recommended 
by  the  American  Society  of  Civil  Engineers: 

Portland  Cement 

Definition:  This  term  is  applied  to  the  finely  pulverized  product 
resulting  from  the  calcination  to  incipient  fusion  of  an  intimate 
mixture  of  properly  proportioned  argillaceous  and  calcareous  ma- 
terials, and  to  which  no  addition  greater  than  3%  has  been  made 
subsequent  to  calcination. 

Specific  Gravity 

The  specific  gravity  of  cement  shall  not  be  less  than  3.10.  Should 
the  test  of  cement  as  received  fall  below  this  requirement,  a  second 
test  may  be  made  upon  a  sample  ignited  at  a  low  red  heat.  The 
loss  in  weight  of  the  ignited  cement  shall  not  exceed  4%. 

Fineness 

It  shall  leave  by  weight  a  residue  of  not  more  than  8%  on  the 
No.  100,  and  not  more  than  25%  on  the  No.  200  sieve. 

Time  of  Setting 

It  shall  not  develop  initial  set  in  less  than  thirty  minutes;  and 
must  develop  hard  set  in  not  less  than  one  hour,  nor  more  than  ten 

Tensile  Strength 

The  minimum  requirements  for  tensile  strength  for  briquettes 
one  square  inch  in  cross  section  shall  be  as  follows  and  the  cement 
shall  show  no  retrogression  in  strength  within  the  periods  specified. 

Neat  Cement 
Age  Strength 

24  hours  in  moist  air 175  Ibs. 

7  days  (1  day  in  moist  air,     6  days  in  water) 500  Ibs. 

28  days  (1  day  in  moist  air,  27  days  in  water) 600  Ibs. 

One  Part  Cement,    Three  Parts  Standard  Ottawa  Sand. 

7  days  (1  day  in  moist  air,     6  days  in  water).. 200  Ibs. 

28  days  (1  day  in  moist  air,  27  days  in  water). 275  Ibs. 


TESTS    OF    CEMENT 


Constancy  of  Volume 

Pats  of  neat  cement  about  three  inches  in  diameter,  one-half 
inch  thick  at  the  center,  and  tapering  to  a  thin  edge,  shall  be  kept  in 
moist  air  for  a  period  of  twenty-four  hours. 

(a)  A  pat  is  then  kept  in  air  at  normal  temperature  and  ob- 
served at  intervals  for  at  least  28  days. 

(6)  Another  pat  is  kept  in  water  maintained  as  near  70°  F.  as 
practicable,  and  observed  at  intervals  for  at  least  28  days. 

(c)  A  third  pat  is  exposed  in  any  convenient  way  in  an  atmos- 
phere of  steam,  above  boiling  water,  in  a  loosely  closed  vessel  for 
five  hours. 

These  pats,  to  satisfactorily  pass  the  requirements,  shall  remain 
firm  and  hard  and  show  no  signs  of  distortion,  checking,  cracking, 
or  disintegrating. 

Sulphuric  Acid  and  Magnesia 

The  cement  shall  not  contain  more  than  1.75%  of  anhydrous 
sulphuric  acid  (SO3),  nor  more  than  4%  of  magnesia  (MgO). 

4.     Quick  Tests 

The  preceding  specifications  ancl  methods  of  investigation 
presuppose  the  conveniences  of  a  testing  laboratory  to  be  at  hand. 
The  constructor,  when  he  comes  upon  a  job,  is  frequently  without 
such  conveniences.  He  is  frequently  compelled  to  decide  whether 
the  cement  is  suitable  or  not  by  employing  such  rough  and  ready 
means  only  as  are  at  hand,  and  a  few  words  as  to  such  practical 
methods  of  investigation  as  must  be  used  are  in  order  for  his  benefit. 

Fineness 

The  constructor  can  readily  determine  whether  the  grinding  is 
reasonably  coarse  or  not  by  feeling  of  a  sample  between  the  thumb 
and  finger  without  recourse  to  screens  or  sieves. 

The  effect  of  fine  grinding  upon  the  cement  is  to  render  samples 
of  mortar  made  of  sand  and  cement  stronger.  In  other  words, 
it  gives  the  cement  a  greater  sand  carrying  power;  it  renders  it  quicker 
setting;  a  stronger  concrete  is  obtained,  or  a  larger  proportion  of 
sand  can  be  used  with  finely  than  with  coarsely  ground  cement 
with  the  same  resulting  strength. 

In  making  briquettes  of  neat  cement,  however,  the  coarsely 
ground  cement  may  show  higher  results,  but  what  the  constructor 
is  interested  in  is  the  result  obtained  with  the  mortar  paste  of  sand 
and  cement  in  the  usual  proportions. 


10  METHOD    OF    MAKING    QUIOK    TESTS 

Accelerated  Test 

The  object  of  this  test  is  to  bring  out  and  make  evident  those 
qualities  which  tend  to  destroy  the  strength  and  durability  of  a  ce- 
ment. As  it  is  highly  essential  to  determine  such  qualities  at  once, 
tests  of  this  character  are  for  the  most  part  made  in  a  very  short 
time,  and  are  known,  therefore,  as  accelerated  tests.  Failure  is 
revealed  by  cracking,  checking,  swelling  or  disintegration,  or  all  of 
these  phenomena.  A  cement  which  remains  perfectly  sound  is 
said  to  be  of  Constant  Volume. 

Failure  to  meet  the  requirements  of  the  accelerated  tests  in  ship- 
ments direct  from  mill  need  not  be  sufficient  ground  for  rejection. 
The  cement  may  be  held  for  twenty-eight  days  and  a  re-test  made 
at  the  end  of  that  period.  But  failure  to  meet  the  requirements  at 
this  time  should  be  considered  sufficient  cause  for  rejection. 

The  accelerated  test  is  a  rough  and  ready  means  for  determining 
without  elaborate  equipment  whether  cement  is  fit  to  use.  Cement 
known  to  have  been  stored  by  a  dealer  for  some  time  should  be 
promptly  rejected  if  it  fails  in  this  test. 

If  a  Portland  Cement  passes  the  accelerated  test  it  may  be  used 
immediately  with  reasonable  certainty  as  to  its  ultimate  soundness. 

Method  of  Testing 

The  method  of  making  the  accelerated  test,  is  as  follows:  On 
a  piece  of  glass  about  four  inches  square,  take  a  sample  of  the  cement 
and  mix  it  to  a  consistency  such  that  the  cement  can  be  readily 
kneaded  without  crumbling  and  at  the  same  time  not  so  soft  as  to 
run  or  lose  its  shape  whan  pressed  into  a  smooth  pat  with  a  thin 
edge.  Place  the  pat  so  formed  under  a  moist  cloth  for  a  period  of 
twenty  four  hours  in  a  temperature  from  sixty  to  seventy  five  degrees 
and  then  expose  it  to  an  atmosphere  of  steam.  Or,  if  preferred, 
the  specimen  after  curing  as  above  for  twenty  four  hours  may  be 
placed  in  cold  water,  which  is  raised  to  and  maintained  at  the  boiling 
point  for  several  hours.  Three  to  four  hours  is  the  usual  period. 
Under  this  test  the  pat  should  harden  without  cracking  or  swelling. 

Causes  of  Unsoundness 

Cracking,  crumbling,  or  disintegration  of  work  in  Portland  Ce- 
ment concrete  properly  mixed  and  laid  may  be  caused  by  an  excess 
of  lime;  by  under  burning  or  by  an  excess  of  magnesia  in  a  thoroly 
burned  cement,  producing  gradual  expansion  which  will  disintegrate 
the  mortar  or  concrete  even  after  several  years. 


TESTING    SAND,    ETC.  11 

Care  of  Cement 

The  inspector  should  see  that  the  cement  is  properly  housed  when 
delivered  to  the  job  and  protected  from  the  elements  so  that  it  will 
not  be  damaged  by  moisture.  Dampness  from  insufficient  protec- 
tion will  render  the  cement  lumpy  and  while  it  may  not  destroy  its 
setting  properties  it  will  greatly  reduce  its  sand  carrying  power 
and  efficiency  or  may  even  render  it  entirely  worthless. 

5.     Specification  for  Aggregate 

Sand:  Sand  used  should  be  clean  and  coarse,  or  a  mixture  of 
coarse  and  fine  grains  with  coarse  grains  predominating,  which  should 
be  free  from  clay,  loam,  mica  and  other  impurities. 

Testing  Sand:  In  order  to  determine  the  amount  of  clay,  dirt 
or  other  impurities,  a  simple,  practical  test  is  to  take  an  ordinary 
quart  glass  preserve  jar,  put  in  a  pint  of  sand,  fill  with  water  and 
put  on  the  cap.  Shake  thoroughly  and  allow  it  to  settle.  The 
result  will  be  that  the  coarser  grains  will  go  to  the  bottom  in  the  order 
of  their  size,  and  the  silt  and  light  impurities  will  settle  in  a  layer  at 
the  top,  giving  the  observer  a  means  of  gaging  the  amount  of  the 
impurities  accurately  and  judging  of  the  character  of  the  sand  and  the 
proportion  of  coarse,  medium  and  fine  grains  in  its  make-up.  From 
three  and  one-half  to  four  percent  of  clay  in  the  form  of  finely  divided 
silt  will  do  no  harm  in  a  bank  sand  or  gravel  for  reinforced  concrete 
work.  Even  higher  percentages  than  this  have  been  claimed  to 
increase  the  strength  of  the  concrete  under  test,  though  where  it 
is  exposed  to  the  elements  and  the  action  of  frost  a  percentage  even 
as  high  as  this  seems  to  be  quite  detrimental.  However,  in  building 
work,  which  is  usually  under  cover,  it  does  no  harm  whatever. 

The  effect  of  the  size  of  the  grains  of  sand  has  been  investigated 
by  Feret.  The  accompanying  figure  from  Johnson's  Materials  of 
Construction,  shows  results  obtained  by  Feret  on  a  1  :  3  mortar 
after  hardening  one  year  in  fresh  water.  The  sand  used  consisted 
of  various  proportions  of  fine  grains  up  to  .5  mm,  medium  .5  to  2mm, 
and  coarse  2  to  5mm,  and  in  the  diagram  the  strength  of  the  mortar 
is  recorded  in  the  triangle  at  such  distances  from  the  base  line  as 
represent  the  proportions  of  each  size  of  sand  used,  the  line  of  equal 
strength  being  wherever  drawn  in  the  diagram.  Thus  the  strength 
of  the  mortar  in  which  only  fine  sand  was  used  was  only  1400  pounds 
per  square  inch.  The  maximum  strength  of  3500  pounds  per  square 
inch  was  obtained  from  a  mixture  containing  85  percent  of  coarse 
sand  and  15  percent  of  fine  with  a  very  little  sand  of  medium  size. 


12 


STRENGTH  OF  MORTAR  DEPENDENT  ON  GRADE  OF  SAND 


Any  point  of  an  entire  contour  line  represents  a  sand  made  up  of 
the  different  sizes  G,  M,  and  F  in  proportions  corresponding  to  its 
perpendicular  distance  from  the  sides  opposite  each  apex  but  having 
the  same  strength  as  every  other  point  on  the  same  line.  This  dia- 
gram shows  that  a  considerable  variation  in  the  proportion  of  coarse 
and  fine  grains  will  make  a  mortar  of  the  same  strength,  but  that, 
in  general,  the  strength  of  a  mortar  with  fine  sand  of  uniform  size 
of  grains  is  about  one  half  or  less  than  one  half  that  of  a  mortar 
made  with  the  same  proportion  of  sand  with  grains  ranging  from 
coarse  to  fine,  and  that  in  general  the  strongest  mortar  is  secured 
with  a  coarse  sand  with  grains  ranging  from  coarse  to  medium. 


—Showing  the  Method  of  Representing 
Proportionate  Mixtures  of  Three  Ingredients. 
G  =  coarse  sand,  0.2  in.  to  0.08  in.  in  diameter. 
M  =  medium*  sand,  008  in.  to  0.0:2  in.  in  diameter. 
F  —  fine  sand  less  than  0.02  in.  in  diameter. 


— Compressive  Resistance  of  Portland- 
cement  Mortars,  in  pounds  per  square  inch.  aft<-r 
nine  months  in  air  and  then  three  months  in 
sea-water.  Mortar  1  0.  :  3  S.  in  all  cas«>s.  but 
the  composition  of  the  sand  varying  according 
I  >  position  hi  the  triangle. 


—  C  nnpressive    Resistance    of  Portland-  —Compressive  Resistance  of  Portland 

cement  Mortars,  1  C. :  3  S  ,  in  pounds  per  square  cement  Mortars,  1  C.  :  3  S.,  in  pounds  per  square 

inch,    after   one    year   in' sea-water.      Shaded  inch,  after  one  year  in  fresh  water. 

Sart  indicates  mixtures   which   were  partially 
Ujntegrated-- 


GRAVEL    AND    BROKEN    STONE  13 

The  effect  of  an  excess  of  clay,  such  for  instance  as  the  dust  from 
soft  magnesium  limestone  will  sometimes  greatly  retard  the  harden- 
ing of  cement,  the  writer  having  seen  instances  where  concrete  at 
the  age  of  a  month  had  not  attained  twenty  five  percent  of  its  normal 
strength  and  where  the  ultimate  strength  was  reduced  perhaps  not 
more  than  twenty  five  or  thirty  percent  by  the  use  of  this  improper 
mixture. 

In  many  specifications,  clean  sharp  sand  is  called  for  in  spite  of 
the  fact  that  in  many  parts  of  the  country  sharp  sand  is  not  obtain- 
able. Sand  with  rounded  grains  such  as  lake  or  beech  sand  is  per- 
fectly satisfactory,  there  being  little  difference  in  strength  between 
the  mortar  made  with  sand  of  angular  or  sharp  grains  and  that 
with  rounded  grains.  The  idea  that  a  sharp,  angular  aggregate  is 
necessary  for  strong  concrete  is  the  basis  for  the  objection  made  by 
some  to  lake  or  bank  gravel  as  a  coarse  aggregate,  while  as  a  matter 
of  fact  better  results  and  stronger  concrete  is  generally  secured  with 
the  round  pebble  than  with  angular  stone,  provided  the  specimen 
tested  is  not  less  than  six  months  old. 

Gravel  where  used,  should  be  composed  of  clean,  hard  pebbles 
and  sand  free  from  clay  and  other  foreign  matter,  such  as  rotten  stone, 
hardened  lumps  of  clay  and  the  like.  A  sample  having  the  coarser 
materials  screened  out  may  be  tested  for  impurities  in  the  same 
manner  as  was  given  for  the  sand. 

Broken  Stone:  Broken  stone  used  should  consist  of  sound 
crushed  stone,  such  as  trap  rock,  limestone,  granite,  hard  sand- 
stone or  conglomerate.  If  the  texture  is  crystalline  and  there  are 
no  portions  of  rotten  stone  or  hardened  clay  such  as  is  sometimes 
found  in  oolitic  limestone  and  shale,  the  crusher  run  may  be  used, 
if  a  part  of  the  sand  content  which  would  otherwise  be  used  in  the 
mix  be  left  out  equivalent  to  the  fine  particles  of  crushed  stone  in 
the  crusher  run. 

If,  however,  the  stone  can  be  readily  reduced  under  the  hammer 
to  a  fine,  impalpable  powder  as  is  the  case  with  some  shales  and 
with  the  oolitic  type  of  limestone  referred  to,  the  dust  should  be 
entirely  lemoved. 

It  is  better,  where  possible  to  use  only  that  stone  which  is  found 
durable  when  exposed  to  the  action  of  the  elements  and  frosts,  and 
the  harder  the  stone  the  stronger  the  concrete  that  may  be  made  when 
using  it  as  the  aggregate. 


14  PROPORTION  OF  MORTAR  TO  COARSE  AGGREGATE 

6.     Proportions  of  Materials 

In  concrete  building  construction  the  proportions  which  exper- 
ience indicates  most  economical  in  concrete  for  slab  and  beam 
construction,  columns  and  column  footings  except  for  the  case  where 
the  loads  to  be  carried  are  unusual^  gieat,  is  one  part  of  cement  and 
two  parts  sand  with  four  parts  broken  stone  or  gravel,  this  being 
indicated  by  the  expression  1  :  2:  4.  These  proportions  are  cus- 
tomarily taken  by  measure,  each  bag  of  cement  being  estimated  as 
equal  to  one  cubic  foot  in  volume,  thus  the  proportions  of  1:  2:  4 
mean  one  sack  of  cement  (94  pounds)  two  cubic  feet  of  sand  and  four 
cubic  feet  of  crushed  stone.  Too  frequently  the  inexperienced  builder 
interprets  a  1  :  2  :  4  concrete  to  mean  a  1  :  (3  aggregate  or  six  parts  of 
gravel  which  may  be  two  thirds  sand  and  one  third  coarse  aggregate. 
The  proportions  of  1  :  2  refer  to  the  mortar  and  mean  the  ratio  of 
the  cement  to  the  volume  of  sand  which  runs  from  about  one  eighth 
inch  down  in  size,  while  coarser  material  than  this  may  be  considered 
as  coarse  aggregate  or  the  stone  content. 

The  size  of  the  stone  in  reinforced  concrete  work  in  ordinary 
building  construction  should  range  from  one  inch  down,  observing 
directions  for  screening  as  detailed  under  the  specification  for  broken 
stone. 

The  first  requirement  in  proportioning  the  aggregate  for  rein- 
forced concrete  vork  is  to  see  that  there  is  an  excess  of  fine  material 
over  and  above  that  required  to  fill  the  voids  in  the  coarse  com- 
ponent of  the  aggregate.  The  volume  of  voids  in  coarse  aggregate 
is  greater  with  a  uniform  size  of  stone  than  where  the  sizes  of  coarse 
aggregate  vary  from  coarse  to  fine,  and  for  that  reason  the  crusher 
run  of  stone  is  preferable  where  the  stone  is  granite,  trap  or  hard 
crystalline  stone.  In  heavy  mass  work,  however,  a  larger  proportion 
of  coarse  aggregate  with  the  size  of  stone  varying  from  three  to  four 
inches  in  diameter  down  can  be  advantageously  used,  but  this  is 
unsuited  for  reinforced  concrete  work  in  the  ordinary  building  line. 
It  is  only  suitable  for  bridge  piers,  and  heavy  mass  work. 

7.     Analysis  of  Strength  of  Concrete 

Concrete  may  be  defined  as  an  artificial  conglomerate  stone  in 
which  the  coarse  aggregate  or  space  filler  (generally  a  hard  natural 
stone,  furnace  slag  or  pebble)  is  held  together  by  a  cement  mortar 
matrix.  Having  selected  a  given  coarse  aggregate,  the  strength 
of  the  concrete  depends  on  the  strength  of  the  mortar  matrix,  in 
other  words,  on  the  ratio  of  cement  to  sand  in  the  mortar  for  all 
samples  of  the  same  age,  formed  under  the  same  conditions. 


ANALYSIS   OF   STRENGTH   OF    CONCRETE  15 

The  Strength  of  the  Concrete  depends  then: 

First,  on  the  grade  of  sand  and  the  proportion  of  the  cement  to 
the  sand  in  the  mortar . 

Second,  upon  the  hardness  and  the  character  of  the  coarse  aggre- 
gate. 

Third,  on  manipulation  and  the  conditions  under  which  the 
concrete  is  cured  or  hardened. 

Fourth,  on  the  age  of  the  specimen. 

Mortar  made  with  a  very  fine  sand  is  only  about  half  as  strong 
as  that  made  with  coarse  and  medium  grains  and  for  that  reason 
the  specification  regarding  the  character  of  the  sand  should  be  given 
careful  attention. 

As  shown  by  Feret,  quite  a  variation  in  the  proportion  of  medium, 
coarse  and  fine  grains  of  sand  will  give  nearly  the  same  strength  so 
that  the  average  clean  coarse  bank  sand  will  generally  fill  the  re- 
quirements for  a  good  concrete  mortar. 

The  richer  the  mortar  the  stronger  the  concrete.  As  noted  above, 
we  recommend  a  one-to-two  mortar  for  reinforced  concrete,  and  where 
high  working  stresses  are  to  be  used  in  reinforced  concrete  columns 
we  would  recommend  a  mortar  in  the  concrete  as  rich  as  one  cement 
to  one  and  one  half  sand,  obtaining  thereby  an  increase  of  twenty 
five  percent  in  the  permissible  working  stress. 

Coarse  Aggregate 

The  effect  of  the  strength  of  the  coarse  aggregate  upon  the  strength 
of  the  concrete,  in  tests  of  concrete  made  with  shale  rock  crushed 
to  one  and  one  half  inches  or  under,  at  Duluth,  show  the  shale  concrete 
about  sixty-five  to  seventy  percent  as  strong  as  trap  rock  concrete 
and  the  trap  rock  concrete  from  ninety  to  ninety-five  percent  as 
strong  as  that  made  with  lake  gravel  for  the  coarse  aggregate.  These 
tests  were  made  on  concrete  about  four  months  old. 

Manipulation  and  Conditions  of  Curing 

While  the  quality  of  the  cement,  sand  and  aggregate  have  more 
or  less  influence  on  the  resulting  concrete,  with  any  good  brand 
of  first  class  Portland  Cement,  clean  coarse  sand  and  hard  crushed 
stone,  substantially  the  same  results  will  be  secured  under  identical 
conditions  of  mixing  and  curing.  The  latter  conditions  have  a 
most  decided  influence  on  the  strength  of  the  concrete,  viz.,  whether 
sufficient  water  has  been  used  to  permit  and  promote  perfect  crystalli- 
zation of  the  cement,  whether  an  excess  amount  of  water  has  been 


16  HARDENING    OF    CEMENT 

used  and  the  fine  and  coarse  materials  have  been  allowed  to  separate 
or  become  segregated,  whether  the  concrete  has  been  thoroly 
mixed,  and  whether  the  conditions  for  curing  were  favorable,  such 
as  keeping  the  concrete  damp  and  preventing  it  from  drying  out 
too  rapidly  or  whether  it  was  hardened  under  unfavorable  cir- 
cumstances in  frosty  weather.  On  this  account  it  is  difficult  to  har- 
monize the  large  number  of  isolated  tests  that  have  been  made 
by  independent  investigators  under  widely  varying  conditions. 

In  building  work,  however,  it  is  a  fortunate  fact  that  except  in 
cold  weather  where  the  work  requires  special  treatment  the  general 
conditions  for  hardening  are  most  favorable.  After  one  floor  has 
been  poured  the  next  is  erected  thereon  within  a  week  or  so  and  the 
excess  water  dropping  from  the  upper  floors  keeps  the  concrete  in 
the  lower  properly  wet,  rendering  the  conditions  more  favorable  for 
hardening  and  curing  than  those  of  the  ordinary  laboratory  test. 

8.     Hardening  of  Portland  Cement. 

The  hardening  of  Portland  Cement  is  a  chemical  process  which 
wTithin  certain  limits  is  accelerated  by  heat  and  retarded  by  cold. 
This  is  an  important  consideration  for  the  builder  to  keep  in  mind, 
since  when  the  temperature  of  the  water  used  in  the  mix  and  the 
aggregate  is  at  or  approximately  near  the  freezing  point,  the  cement 
lies  dormant  and  no  fixed  rule  can  be  given  of  a  set  number  of  days 
during  which  time  it  is  necessary  for  the  concrete  to  lie  in  place  on 
the  forms  before  it  will  attain  a  certain  given  degree  of  strength. 
In  hardening,  as  in  nearly  all  chemical  reactions,  heat  is  generated 
by  the  setting  of  the  cement.  This  heat  is  radiated  away  very 
rapidly  where  the  mass  is  small  or  thickness  of  the  part  of  the  con- 
crete work  is  inconsiderable,  while  where  the  mass  is  large,  as  in  the 
case  of  heavy  walls,  piers,  and  the  like,  the  heat  generated  by  the 
setting  of  the  cement  is  not  lost  rapidly  by  radiation  and  the  work 
tends  to  cure  much  more  rapidly  in  heavy  work  in  cold  weather  than 
in  the  case  of  the  thin  slabs  usually  used  in  the  floors  of  buildings. 
Special  directions  for  the  treatment  of  concrete  at  various  tempera- 
tures will  be  given  under  a  special  section  dealing  with  building 
work. 

Increase  of  Strength  of  Concrete  with  Age 

The  following  table  shows  compressive  strength  of  concrete 
as  determined  by  test  made  at  the  Watertown  Arsenal  in  1899. 
1:2:4  mixture. 


COEFFICIENT    OF    EXPANSION  17 


Brand  of  cement 

7  days 

1  month 

3  months 

6  months 

Atlas.... 

1,387 

2,428 

2,966 

3,953 

Alpa... 

904 

2,420 

3,123 

4,411 

t^ 

Germania 

2,219 

2,642 

3,082 

3,643 

Alsen... 

1.592 

2.269 

2.  60S 

2  612 

Average...  1,525  2,240  2,944  3,904 

The  above  gives  a  fair  idea  of  the  increase  in  strength  of  concrete 
with  age  under  normal  temperature  above  60°  F. 

After  a  period  of  six  months  the  concrete  in  ordinary  building  is 
found  to  increase  slowly  in  strength  and  considerably  in  hardness 
and  rigidity.  Thus  it  appears  that  the  stiffness  of  a  long  span  slab 
will  increase  about  twenty  percent  between  two  months  and  twelve 
to  fifteen  months,  and  the  strength  perhaps  in  a  lesser  ratio,  on  the 
assumption  that  the  compression  element  only  in  the  combination 
is  hardening  and  increasing  in  strength. 

Coefficient  of  Expansion 

The  coefficient  of  expansion  of  concrete  is  practically  the  same  as 
that  of  mild  steel.  Some  investigators  have  made  this  coefficient 
per  degree  of  Fahrenheit  slightly  less  and  others  slightly  more  than 
.0000065,  which  is  usually  accepted  for  mild  steel,  hence  ordinary 
changes  of  temperature  cause  no  injury  to  the  composite  material 
formed  by  embedding  steel  in  concrete. 

9.     Bond  between  Concrete  and  Steel. 

In  the  design  of  any  combination  of  concrete  with  steel  the  bond 
between  the  two  elements  is  of  prime  importance.  Concrete  setting 
in  the  air  shrinks  and  grips  the  reinforced  members  with  a  vice-like 
grip.  The  richer  the  mixture  the  greater  this  shrinkage  stress  and 
the  better  the  bond.  In  concrete  setting  in  water  this  shinkage 
is  lacking  and  in  this  case  deformed  reinforcement  or  mechanical 
bond  is  desirable. 

The  bond  between  the  concrete  and  steel  has  a  maximum  value 
with  a  plastic  mix  of  concrete  such  that  the  mortar  will  flow  slowly 
and  thoroly  surround  the  metal.  It  is  greatly  reduced  with  a 
stiff  mixture  requiring  tamping  and  at  the  other  extreme  also  is  less 
with  too  sloppy  a  mixture  of  concrete. 

With  plain  round  rods  embedded  12  inches  the  bond  value  may 
reach,  under  favorable  conditions,  a  maximum  of  600  pounds  per 
square  inch  with  concrete  of  a  1  :  2  mortar,  six  months  old,  but 


18 


ADHESION    OR    BOND 


with  dry  tamped  concrete  the  bond  value  or  adhesion  as  it  is  some- 
times called  may  run  as  low  as  200  pounds  per  square  inch  of  the 
surface  of  the  bars. 

A  round  bar  will  give  a  higher  bond  value  than  a  flat  or  rec- 
tangular shape,  while  a  polished  or  cold  rolled  shaft  as  it  comes  from 
the  mill  will  give  a  value  hardly  more  than  a  fourth  as  great  as  that. 
Slight  rusting  of  the  surface  improves  the  adhesion  or  bond  since  the 
rust  combines  chemically  with  the  cement  and  seems  to  increase  the 
shrinkage  grip.  Further,  slight  rusting  tends  to  remove  the  black 
mill  scale  making  the  adhesion  uniform  along  the  surface  of  the 
metal. 

Paint,  oil  or  grease,  greatly  reduces  the  adhensioii.  With  properly 
arranged  reinforcement  the  designer  rarely  has  occasion  to  figure 
upon  the  bond  value  between  the  two  materials,  since  it  is  amply 
provided  for  where  due  precaution  has  been  taken  to  render  the 
design  safe  to  execute  by  properly  tying  the  materials  together  by 
an  ample  lap  of  the  metal  over  the  supports  and  the  use  of  sufficient 
cement. 

Occasionally  the  designer  may  be  forced  to  use  short  stock  lengths 
in  beams  or  slabs,  and  under  such  conditions  a  working  stress  not 
exceeding  forty  pounds  per  square  inch  is  permissible  providing  the 
bars  are  also  hooked  at  the  ends.  Even  with  this  additional  precau- 
tion care  must  be  exercised  in  keeping  the  work  supported  much 
longer  than  would  be  necessary  with  preferable  lengths  of  rods. 
The  reason  for  this  precaution  and  the  low  value  recommended  is 
that  the  bond  strength  between  concrete  and  steel  varies  greatly 
with  trie  age  of  the  concrete  and  like  the  shearing  resistance  it  is 
very  low  with  partly  cured  concrete  but  increases  with  the  age  and 
hardness  of  the  work,  tho  less  rapidly  than  the  resistance  in  com- 
pression. 

The  constructor  should  keep  clearly  in  mind  the  important 
conditions  which  insure  satisfactory  bond  between  the  metal  and 
concrete,  namely:  A  mixture  of  proper  consistency  containing 
sufficient  cement  and  the  preferable  use  of  bars  round  in  section, 
which  are  most  readily  surrounded  by  the  plastic  concrete  in  flowing. 

It  is  interesting  to  note  in  the  following  table  of  tests  of  bond 
that  the  average  adhesion  of  the  \"  round  rods  was  twenty  percent 
more  than  that  of  the  square  rods  with  the  1  :  3  mortar;  that  the 
adhesion  to  the  steel  with  the  broken  stone  concrete  was  greater 
than  with  the  1  :  3  mortar  or  even  the  neat  cement  test;  and  that 
the  adhesion  to  the  quarter  inch  by  one  inch  averaged  only  six 
tenths  that  of  the  half  inch  rounds. 


BOND 


19 


TABLE  I 

SHOWING    ADHESION  OF  VARIOUS  SHAPED    RODS  TO  CONCRETE 
OF  CONSTANT  COMPOSITION 

1  Part  Cement  to  3  Parts  Sand 

*9-10-ll-12,  1  Cement,  2  Sand,  4  Broken  Stone 

(Even  numbers  40  days — Odd  Numbers  80  Days 


No. 

Section  of       Length 
Steel           Embed- 

Perimeter, 
ins. 

Load  in 
Ibs.  at 

Area  of 
contract 

Load  in 
Ibs.  per 

Average 
per 

ment  ins. 

Failure 

sq.  ins. 

sq.  ins. 

sq.  in. 

1  ] 

6* 

1      

12.25 

2    1 
3 

\"  square'.  .  . 

6!              2.0 
6               

5,700 
5,200 

13.00 
12.00 

438 
433 

432 

4 

6|             5,300 

12.50 

424 

5 

6               4,600 

9.43 

4SS 

6 

7 

\"  round..  .  . 

6 
6 

1.571 

5,000 
4,600 

9.43 
9.43 

530 

488 

512 

8 

,     5| 

5,000 

9.23 

542 

9 

61 

4.900 

15.63 

313 

10 

i"  v  i  n 

61 

2^5 

4^00 

15.63 

282          293 

11    f 

1      XI      .... 

6| 

4,800 

15.20 

314 

12  j 

61 

4,400 

15.63 

282 

5  } 

f  101 

17,400 

41.0 

424 

6 

10| 

15,800 

40.5 

390 

411 

7 

10* 

17,000 

40.5 

420 

8 
*  9 

I"  square...     |    ™\ 

16,800 
21  200 

41.0 
40.5 

410 
523 

HO 

10i             24,600 

41.0 

600 

587 

*11 

10|             24,200 

40  .  5 

598 

*12 

,   10f 

26,000 

41.5 

627 

Test  by  Emerson,  Eng.  News,  1904,  p.  222. 

That  the  bond  between  the  concrete  and  steel  is  really  a  shrinkage 
grip  may  be  easily  proved  by  the  simple  experiment  of  molding  some 
concrete  and  placing  a  piece  of  round  steel  on  top,  lightly  pressing 
it  into  the  concrete  without  immersing  it  more  than  one  third  the 
diameter.  When  the  concrete  has  cured  it  will  be  found  that  there 
is  very  little  difficulty  in  removing  the  steel.  If  the  piece,  however, 
is  pressed  into  the  concrete  to  a  considerable  depth  and  the  con- 
crete in  its  plastic  condition  allowed  to  flow  around  the  bar  it  will 
be  very  difficult  indeed  to  remove  and  it  will  be  found  that  this  is 
caused  by  the  shrinkage  of  the  concrete  around  the  bar  in  hardening. 
That  this  bond  between  the  concrete  and  the  rod  is  not  due  to  direct 
adhesion  may  be  further  proved  by  splitting  the  concrete  about 
the  bar  or  sawing  it  down  to  the  side  of  the  bar  on  each  side  when 
it  will  be  found  that  the  bar  is  readily  removed  from  the  concrete. 

The  following  table  is  given  by  Professor  Hatt.  The  tests  were 
made  by  drawing  out  the  rods.  In  this  table  it  will  be  noted  that 


20 


BOND 


the  depth  of  the  rod  in  the  concrete  is  much  larger  than  in  the  tests 
by  Emerson,  while  some  tests  by  Feret  with  the  rod  embedded 
2f"  give  values  approximately  half  as  great  as  where  the  length 
embedded  is  from  ten  to  twelve  inches. 

TABLE  II. 


Diameter 

Age  of 

Depth  of 

rnH  in 

Adhesion  in  pounds  per  square 

of  rod 
inches 

specimen 
in  days 

concrete 
inches 

inch  of 

surface  in  contact 

Maximum 

Minimum 

Average 

7/16 

32 

72 

735 

470 

635 

5/8 

35 

76 

780 

714 

756 

The  adhesive  resistance  varies  somewhat  with  the  depth  to 
which  the  rod  is  embedded. 

It  is  greater  with  a  rough  than  with  a  smooth  surface. 

It  increases  with  the  proportion  of  cement  up  to  a  certain  limit. 

It  is  a  maximum  with  a  plastic  mix  and  a  minimum  with  a  dry 
mix  and  tamped  concrete. 

It  increases  with  the  age  of  the  concrete. 

Considere  finds  that  for  concrete  exposed  to  air  the  amount  of 
water  used  in  mixing  has  a  great  influence,  too  dry  concrete  adhering 
badly.  An  excess  of  water  giving  the  concrete  the  necessary  fluidity 
for  fillin'g  up  the  voids  around  the  reinforcement  produced  the  best 
results.  He  considered,  however,  that  this  advantage  of  wet  con- 
crete was  counterbalanced  by  a  notable  diminution  of  tensile  and 
compressive  resistance.  This  would  be  true  were  it  not  a  fact  that 
the  excess  water  in  casting  reinforced  concrete  work  in  building 
construction  is  readily  disposed  of  by  absorption  of  the  forms  and 
leakage  through  them. 

The  low  values  found  by  some  investigators  for  adhesion  or  bond 
seem  to  be  readily  accounted  for  by  the  prevalent  French  custom  of 
tamping  dry  or  stiff  mixtures  of  concrete  rather  than  of  pouring,  if 
of  a  wet  or  plastic  consistancy,  as  is  done  by  the  American  construc- 
tor today.  The  early  idea  was  that  good  concrete  could  only  be  made 
by  a  dry  mix  and  tamping.  Combined  with  this  dry  mix  the  deformed 
bar  was  unquestionably  an  improvement,  but  as  the  use  of  the  dry 
mix  has  long  been  abandoned  the  main  advantage  in  the  use  of  the 
deformed  bar  has  largely  disappeared  with  its  abandonment. 


VARIATION    OF    STRENGTH    WITH    MOISTURE  21 

By  far  the  greater  number  of  concrete  failures  have  occurred 
where  deformed  bar  reinforcement  has  been  used.  This  is  due 
in  part  probably  to  the  fact  that  some  types  of  deformed  bar  re- 
inforcement are  such  that  with  ordinary  care  the  metal  is  not  so 
well  surrounded  as  in  the  case  with  plain  rounds,  and  the  shrink- 
age grip  of  adhesion  is  less  with  the  irregular  section.  Further,  the 
designer  seems  frequently  to  place  too  great  confidence  in  the  bond 
strength  in  these  designs,  and  has  frequently  neglected  a  sufficient 
lap  over  the  support,  to  render  the  design  safe  and  conservative. 

Plain  reinforcement  would  no  doubt  not  have  done  much  better 
than  deformed  bar  reinforcement  with  the  same  arrangement  and 
and  length  of  lap,  but  over  confidence  in  the  deformed  bar  reinforce- 
ment has  had  a  tendency  to  lead  to  a  type  design  inherently  dangerous 
with  any  type  of  reinforcement. 

10.     Variation  in  the  Strength  of  Concrete  with  Variation 
of  Temperature  and  Moisture. 

This  question  is  of  the  greatest  importance  to  the  constructor  in 
putting  up  work.  The  concrete,  partly  cured,  may  apparently  be 
stiff  and  rigid  when  the  forms  are  removed  in  cold  or  freezing  weather. 
Then  with  a  sudden  change  in  the  temperature,  such  as  may  readily 
be  brought  about  by  putting  a  heating  plant  into  the  building,  the 
concrete  will  sweat  and  soften  and  get  out  of  shape.  Again,  when 
concrete  which  has  had  as  long  as  two  to  three  months  in  which  to 
cure  during  the  fall  season,  is  exposed  all  winter,  soaked  with  water 
and  the  water  frozen,  and  in  the  spring  a  heating  plant  is  put  into 
the  building  and  the  slab  thawed  out,  its  strength  is  temporarily 
greatly  reduced,  and  if  the  slab  is  carrying  the  weight  of  other  stories 
which  are  shored  from  it,  permanent  deflection  and  serious  trouble 
may  result. 

The  older  and  more  thoroly  cured  the  concrete,  the  more 
rigid  it  is  and  the  less  the  variation  in  strength  resulting  from  the 
conditions  above  noted.  Concrete  which  is  thoroly  soaked  with 
water  is  less  rigid  in  compression  than  concrete  which  is  thoroly 
dried  out.  This  change  in  strength  is  due  to  the  fact  that  the  con- 
crete expands  with  moisture  and  shrinks  or  contracts  as  it  dries  out, 
this  action  being  greater  with  concrete  during  the  hardening  stages. 

Hence,  the  constructor  should  use  care  and  see  that  his  work  is 
not  over-loaded,  particularly  at  the  time  when  he  is  firing  up  the  heat- 
ing plant  in  a  building  in  which  the  floors,  though  they  have  had 
some  time  to  cure  in  the  fall,  have  been  thoroly  soaked  and  frozen. 


22  MIXING 

Undue  confidence  in  the  strength  of  partly  cured  and  frozen  concrete 
arises  from  observing  that  when  the  forms  are  first  removed  there 
is  no  deflection  and  the  concrete  stands  up  apparently  of  ample 
strength  and  rigidity  under  the  superimposed  load  of  the  centering 
of  one  or  more  stories  above.  A  sudden  change  in  temperature, 
causing  the  moisture  in  the  slab  to  thaw  and  expand  in  the  concrete 
will  so  weaken  the  slab  that  a  large  permanent  set  will  result. 
The  cautious  builder  will  keep  the  floor  well  shored  up  until  he  is 
sure  that  it  is  thoroly  dried  out  by  heat  so  that  there  is  no  chance 
for  the  work  to  get  out  of  shape  as  above  explained. 

The  extent  to  which  the  strength  of  a  slab  may  be  reduced  even 
after  it  has  been  once  fairly  well  cured,  but  subsequently  exposed  to 
the  weather,  soaked  and  frozen  is  illustrated  by  a  case  where  it  was 
so  softened  in  thawing  out  as  to  deflect  eight  times  as  much  as  it 
did  under  identically  the  same  load  after  drying  out  and  exposure  to 
heat  for  three  weeks,  so  that  the  importance  of  the  precaution  above 
outlined  should  be  apparent.  In  this  case  the  slab  was  cast  in  the 
latter  part  of  August,  was  fairly  well  cured  but  exposed  to  the  weather, 
flooded  by  rain  and  frozen  up  during  the  winter.  The  heating  plant 
was  placed  under  the  floor  sometime  in  March  and  a  light  load  then 
placed  on  the  slab  which  had  a  span  of  about  22  feet  and  was  7  inches 
thick,  well  reinforced.  A  deflection  resulted  of  approximately 
2  inches.  The  slab  returned  to  its  original  shape  after  removal  of 
the  load,  and  when  thoroly  dried  out  the  deflection  under  the  same 
load  was  hardly  3/16  inches. 

Deflections  are  found  to  be  increased  where  the  slab  is  wet,  and 
the  strength  is  apparently  somewhat  diminished. 

11.     Machine  Mixing. 

Concrete  for  a  concrete  steel  building  should  be  machine  mixed, 
preferably  in  a  batch  mixer.  Some  of  the  continuous  mixers  do 
good  work  where  bank  gravel  is  used  as  the  aggregate  and  fail  where 
crushed  stone  is  used.  A  batch  mixer  such  as  the  Smith,  Cube, 
Polygonal  or  Ransome,  is  to  be  preferred  because  it  may  be  charged 
with  cement,  sand  and  stone  by  measure  and  the  exact  amount  of 
water  added.  The  water  content  in  the  mix  is  best  supplied  for 
a  large  piece  of  work  by  a  tank  which  will  contain  the  amount  of 
water  needed  for  a  batch  arranged  with  the  usual  float  trap  valve 
so  that  all  the  operator  needs  to  do  is  to  pull  the  string  and  the  tank 
of  water  is  discharged  at  once  in  the  mixer.  This  insures  a  mixture 
of  uniform  consistency  and  effects  a  material  saving  of  time. 


MIXING    PLANT 


23 


Where  the  work  is  of  sufficient  magnitude  to  permit  an  overhead 
hopper  into  which  the  sand  and  stone  may  be  elevated,  and  discharged 
by  gravity  into  the  mixer  as  desired,  a  large  saving  in  labor  is  secured. 
Where  the  mixing  plant  is  near  a  track  the  hopper  may  be  filled  from 
the  cars  by  a  derrick  and  suitable  clam.  Where  the  aggregate  is 
brought  to  the  building  by  team  load,  a  platform  arranged  so  that 
the  wagon  may  be  driven  over  it  and  the  stone  or  sand  dumped  there- 


Fig.  1.     Mixing  Plant,  Lindeke- Warner  Building,  St.  Paul. 

on  and  then  elevated  and  discharged  into  the  top  of  the  hopper,  is 
about  as  economical  an  arrangement  as  the  writer  has  seen. 

A  view  of  a  mixing  plant  of  this  kind  used  in  the  erection  of  the 
Lindeke-Warner  building  of  St.  Paul,  erected  by  Butler  Bros.,  is 
shown  in  Fig.  1. 

In  Fig.  2  is  shown  the  mixing  plant  used  in  the  erection  of  the 
John  Deere  Plow  Company's  building  in  Omaha,  the  hopper  in  this 


24 


POURING    CONCRETE 


case  being  charged  by  the  locomotive  crane,   using  tlje  clam  for 
transferring  the  materials  from  the  cars  to  the  hopper. 

Consistency  of  Concrete 

For  building  construction  and  reinforced  concrete  work  generally 
it  is  necessary  that  the  concrete  shall  be  mixed  so  that  it  will  flow 
slowly  and  thoroly  surround  the  reinforcement  but  it  should 
be  no  more  plastic  than  is  required  to  attain  this  result.  If  mixed 
too  dry  and  tamping  is  depended  upon,  voids  will  be  left  around  the 
steel  and  the  face  of  the  concrete  when  the  forms  are  removed  will 


Fig.  2.      Mixing  Plant  and  Hopper,  John  Deere  Plow  Company  Building,  Omaha. 

be  found  rough  and  full  of  pockets  and  the  work  will  present  an  ap- 
pearance of  weakness  which  it  very  likely  does  not  possess. 

12.     Pouring  Concrete. 

In  pouring  concrete,  the  lowest  portions-  of  the  forms  should  be 
filled  first :  Thus  necessitating  the  least  possible  flow  of  the  concrete 
to  reach  its  final  position  in  the  work.  In  buildings  the  columns 
should  be  filled  first,  then  the  beams  and  finally  the  slabs,  the  opera- 
tion being  continuous  as  far  as  practicable.  If  an  attempt  is  made 
to  reverse  this  program  and  fill  the  beam  before  the  column  is  filled 
the  concrete  will  flow  in  an  inclined  direction  to  the  column  and  as 
each  batch  is  deposited  the  more  liquid  portions  washing  over  the 


JOINTS    IN    THE    WORK  25 

inclined  surface  carry  the  light  laitance  and  fine  sand  down  into  the 
column,  and  an  inferior  concrete  and  one  of  little  strength  will  be 
found  at  the  bottom  on  removal  of  the  column  forms. 

In  pouring  columns,  especially  where  closely  spaced  spiral  hoop- 
ing is  used,  concrete  should  be  poured  over  the  center.  If  an  attempt 
is  made  to  fill  the  column  from  the  side  the  space  between  the  form 
and  hooping  is  filled  up  to  a  considerable  height  in  advance  of  the 
central  core,  the  hooping  acting  as  a  screen  prevents  the  coarse 
aggregate  from  flowing  to  the  lower  central  area  with  the  result 
that  the  mortar  flows  to  the  core  and  seals  to  some  extent  the  voids 
toward  the  hooping  and  the  next  batch  cannot  flow  to  fill  in  the  voids 
thus  left  in  nearly  clean  coarse  aggregate  between  the  hooping  and 
the  column  form.  This  leaves  rough  unsightly  work  when  the  forms 
are  removed  and  while  the  core  may  be  sound  the  fireproofing  is 
inferior  in  character  or  worthless,  and  the  Avork  presents  an  appear- 
ance of  weakness  which  it  does  not  possess. 

Where  to  Make  Joints  in  the  Work  and  How  to  Do  It 
Splicing  of  concrete  in  beams  and  slabs  should  preferably  be  made 
in  the  center  and  should  be  vertical.  The  reason  for  this  is  that 
where  the  concrete  is  allowed  to  flow  out  on  an  inclined  plane  in  the 
beam  the  inert  material  known  as  laitance  comes  to  the  surface, 
preventing  a  good  bond  when  the  new  concrete  is  added.  In  fact 
instances  have  not  infrequently  been  observed  where  wedge  shaped 
pieces  of  concrete  three  feet  long  and  running  from  two  inches  in 
thickness  to  one  quarter  inch  at  the  end  dropped  away  from  the  beam 
owing  to  this  manner  of  placing,  the  bond  being  insufficient  to  carry 
the  weight  of  the  piece.  The  remedy  is  to  break  up  the  surface  of 
the  old  concrete  and  grout  it  with  a  neat  cement  before  proceeding 
to  cast  the  new  work. 

In  very  hot  weather  the  crushed  stone  may  readily  get  so  dry 
and  hot  that  it  absorbs  the  water  from  the  mix  and  causes  the  cement 
to  set  too  rapidly.  When  this  is  the  case  an  open  crack  will  appear  at 
a  joint.  The  remedy  is  to  first  cool  the  stone  by  thoro  wetting 
with  the  hose. 

13.     How  to  Determine  when  Concrete  is  Thoroughly  Cured. 

It  may  be  noted  that  most  failures  occur  in  the  cold  season,  and 
the  need  for  a  certain  and  simple  method  of  determining  whether 
the  concrete  has  been  merely  hardened  by  frost  or  really  cured  is 
obvious.  A  good  way  in  cold  weather  work  is  to  cast  a  few  small 
sample  cubes  or  cylinders  when  pouring  the  floor,  and  allow  them  to 


26  DETERMINING    WHEN    CONCRETE    IS    CURED 

harden  under  the  same  conditions  as  the  slab,  and  these  samples 
may  then  be  examined  at  any  later  time.  When  this  has  not  been 
done,  cut  out  a  small  piece  of  the  concrete  and  place  it  over  a  stove 
or  radiator  and  if  the  concrete  has  been  merely  stiffened  by  frost 
it  will  sweat  and  soften  up,  while  if  cured  it  will  remain  firm,  dry 
and  hard. 

In  the  previous  pages,  dealing  with  the  hardening  of  concrete 
the  fact  has  been  emphasized  that  curing  has  no  direct  relation  to 
the  number  of  days  that  a  floor  has  been  poured.  Hardening  is 
to  be  considered  first  in  connection  with  the  temperature  and  humidity 
during  the  period  of  curing,  and  second  in  connection  with  the  treat- 
ment of  the  materials  in  cold  or  chilly  weather  as  to  whether  the  water 
and  aggregate  had  been  properly  heated. 

In  the  summer  season,  during  ten  or  twelve  days  of  continued 
chilly,  rainy  weather,  concrete  frequently  hardens  so  slowly  that 
the  early  removal  of  the  forms  at  that  period  will  result  in  permanent 
damage  to  the  work,  so  that  the  humidity  and  rain  must  be  taken 
into  consideration  as  well  as  the  temperature  under  which  the  con- 
crete is  cured,  in  deciding  the  feasibility  of  removing  false  work. 

In  the  chilly  weather  of  the  spring  or  fall,  if  the  concrete  has  been 
mixed  with  cold  water  and  subsequently  chilled  by  frost  before  it 
has  had  time  to  set  appreciably  it  may  be  very  slow  in  curing.  It  is 
exceedingly  difficult  under  such  circumstances  for  the  most  expert 
to  tell  when  the  false  work  may  be  removed  and  no  undesirable 
results  follow.  The  concrete  may  after  such  treatment,  (improper 
mixing  with  cold  water  at  a  time  when  the  weather  is  chilly  and 
frosty)  apparently  be  hard  and  subsequently  sweat  and  soften  during 
the  continued  hardening  process  and  the  work  get  out  of  shape. 
That  is,  the  slab  or  beam  may  deflect  permanently  \  or  f ".  Such 
deflection,  while  it  will  not  result  in  permanent  weakness  after  the 
concrete  has  finally  hardened,  will  cause  the  owner  to  question  its 
strength.  It  may  cause  partitions,  which  were  built  upon  it  at  its 
original  elevation,  to  be  left  unsupported  and  a  year  or  a  year  and 
one  half  after  the  inelastic  sag  has  occurred,  the  partitions  will 
commence  to  crack  and  come  down  to  a  bearing.  The  owner  will 
feel  certain  that  the  concrete  work  is  weak,  although  it  has  hardened 
up  and  is  of  ample  stiength,  tho  not  in  the  position  in  which  it 
was  left  when  the  forms  were  removed.  These  troubles  are  entirely 
obviated  by  the  proper  treatment  of  the  materials  in  mixing,  as  we 
have  explained  heretofore. 

As  a  fair  example  showing  the  results  that  may  be  secured  by  the 


CONCRETE    ABOVE    FREEZING  27 

proper  treatment  of  the  concrete,  we  may  cite  the  case  of  a  build- 
ing in  Fort  William,  Ont.,  in  which  the  roof  slab  was  cast  on  the 
seventh  day  of  December,  at  a  temperature  twenty  degrees  below 
zero.  On  the  15th,  a  fire  was  started  by  the  carelessness  of  a  workman 
in  placing  a  kettle  of  pitch  with  which  cork  board  was  being  applied 
too  near  a  salamander  and  the  centering  burned  out  from  under 
this  roof  completely  and  also  from  beneath  a  part  of  the  slab  below. 
Notwithstanding  the  season  of  the  year  and  the  low  temperature  at 
which  the  work  was  cast,  these  slabs  which  were  approximately 
seventeen  feet  in  span,  stood  up  very  well.  The  under  side  of  the 
concrete,  however,  was  somewhat  pitted  by  the  formation  of  steam 
in  small  cavities  and  the  forcing  out  of  small  chunks  of  the  concrete 
thereby.  In  this  case  the  pitting  caused  by  the  fire  was  readity 
remedied  and  the  work  put  in  shape  at  a  comparatively  small  cost. 

14.     Handling  Concrete  Above  Freezing. 

Handling  concrete  to  get  the  best  results  requires  quite  different 
treatment  at  different  times  depending  on  the  temperature.  Per- 
haps the  most  favorable  conditions  under  which  concrete  may  be 
placed  are  at  temperatures  ranging  from  45  degrees  to  50  degrees 
Fahr.  Under  these  conditions  the  concrete  does  not  dry  out  too 
rapidly  and  while  it  may  set  slowly  it  hardens  up  better  than  when 
the  temperature  is  higher. 

In  hot,  dry  weather  the  moisture  dries  out  of  the  concrete  too 
rapidly,  requiring  for  the  best  results,  that  the  work  be  wet  down 
with  a  hose,  particularly  during  the  first  day's  exposure  to  the  sun 
after  casting.  Wetting  should  commence  as  soon  as  the  surface  of 
the  concrete  has  set. 

Frequently  in  the  hot  sun  large  cracks  will  open  up  due  to  the 
rapid  evaporation  of  the  water.  These  can,  and  should  be  promptly 
filled  with  a  bucket  of  grout.  Sometimes  the  stone,  when  exposed 
to  the  heat  of  the  sun  will  become  so  dried  out  and  hot  that  it  will 
absorb  the  water  rapidly  from  the  mix  and  the  heat  of  the  stone  will 
be  sufficient  to  cause  the  concrete  to  set  before  it  can  be  spread  in 
place.  Wetting  the  rock  pile  down  will  eliminate  this  difficulty. 

At  all  temperatures  below  45  degrees  Fahr.,  it  is  best  to  warm 
the  water  and  wake  the  cement  up,  otherwise  it  is  liable  to  set  too 
slowly  to  enable  the  forms  to  be  safely  removed  at  the  usual  intervals 
common  in  warm  weather. 

In  putting  up  a  large  building  at  Toledo,  the  contractor  wired  to 
Minneapolis  for  the  writer  to  visit  the  building,  stating  that  the 


28  CONCRETE  BELOW  FREEZING 

cement  in  the  entire  third  floor  was  not  setting  up  and  had  been  in 
two  weeks.  The  writer  was  unable  to  leave  immediately  and  arrived 
at  the  building  three  days  after  receiving  the  telegram.  The 
weather  had  turned  warm  in  the  meantime  and  the  cement  had 
started  to  set.  There  were  places,  however,  where  a  twenty-penny 
nail  could  be  pushed  into  the  concrete  with  the  thumb  to  its  full 
length.  The  cement  had  been  mixed  with  water  ice  cold,  and  had 
lain  dormant  during  the  chilly  weather  which  succeeded  the  two 
weeks  after  placing  the  concrete.  For  the  remaining  stories,  after 
instructing  the  foreman  to  see  that  the  water  with  which  the 
concrete  was  mixed  was  heated  to  about  120  or  130  degrees  there 
was  no  difficulty  and  the  forms  were  promptly  removed  every  ten 
or  twelve  days  per  story  although  the  weather  was  much  colder  as 
the  season  advanced. 

Concrete  Below  Freezing 

In  freezing  weather  it  is  desirable  to  wake  up  the  cement  by  using 
hot  water.  Water  may  be  heated  to  160  or  180  degrees  and  the  sand 
and  stone  mixed  with  water  in  the  machine  before  adding  the  cement. 
The  water  will  thus  warm  up  the  stone  and  when  the  cement  is  dump- 
ed into  the  mixer  the  temperature  will  probably  be  in  the  neighbor- 
hood of  120  degrees. 

When  the  temperature  is  below  zero,  boiling  hot  water  is  some- 
times used.  The  sand  and  stone  are  first  placed  in  the  mixer,  then 
the  boiling  water  is  added  to  warm  up  the  sand  and  stone,  and 
finally  the  cement,  when  the  sand  and  stone  has  been  warmed  up 
and  the  water  has  been  cooled  down  to  a  temperature  of  not  more 
than  120°.  This  method  of  procedure  is  necessary  in  order  to  not 
kill  the  cement  by  the  boiling  water. 

At  low  temperatures  salt  may  be  advantageously  used.  The 
proportion  of  salt  which  it  is  desirable  to  use  for  temperatures  between 
zero  and  25  above  zero  is  a  pint  and  one-half  of  salt  to  each  batch 
containing  two  bags  of  cement.  Below  zero,  a  little  more  than 
a  pint  per  sack,  and  extra  care  is  to  be  taken  in  heating  the  materials 
and  seeing  that  the  concrete  gets  into  place  hot. 

For  these  very  low  temperatures  it  is  much  better  to  heat  the 
stone  and  sand  over  a  coil  of  steam  pipe' if  such  is  available.  In  one 
case  concrete  was  placed  at  28  below  zero  and  the  work  in  this  case 
was  executed  in  a  very  satisfactory  manner.  In  addition  to  the  use  of 
hot  water  the  gravel  used  as  an  aggregate  was  thoroly  heated  over 
a  coil  of  steam  pipes.  There  is  much  less  difficulty  in  placing  concrete 
in  large  masses  in  cold  weather  than  in  thin  slabs  and  the  like. 


PROTECTION    OF    THE    WORK 


29 


In  large  masses  such  as  thick  walls,  thick  slabs,  and  the  like, 
the  cement  generates  heat  in  setting,  sufficient  to  keep  the  body  of 
the  material  warm,  whereas  in  thin  slabs  this  is  not  always  the  case 
and  the  concrete  may  and  frequently  does  freeze. 

It  comes  then  to  a  question  as  to  how  to  handle  the  concrete  in 
the  best  and  most  practical  manner.  In  a  building  having  exterior 
bearing  walls  the  walls  are  built  up,  then  the  slab  is  cast  and  artificial 
heat  should  be  promptly  applied  on  the  under  side  of  the  slab  to 
sweat  out  the  concrete  and  enable  it  to  harden  up  promptly. 
Window  openings  may  be  readily  closed  with  canvas  or  light  cloth. 


Fig.  3.     Canvas  curtain  protecting  an  upper  story  of  the  Strong-Warner  Building, 
St.  Paul,  under  construction  in  winter. 

Where  the  building  is  a  skeleton  concrete  frame  it  should  be  pro- 
tected in  winter  from  outside  temperatures  by  help  of  canvas  curtains 
in  lieu  of  the  walls  as  shown  in  Fig.  3.  Then  the  concrete  may  be 
artifically  heated  and  hardened. 

In  a  large  piece  of  work  the  most  economical  method  of  heating 
is  to  put  in  a  small  fan  with  the  usual  steam  coil,  and  heat  the  building 
by  blowing  in  heated  air  in  the  usual  manner.  In  a  small  building 
this  is  too  expensive  and  the  resort  is  had  to  salamanders  and  coke  for 
heating. 


30  USE    OF    SALT 

15.     Action  of  Salt 

The  action  of  salt  on  concrete  is  three-fold : 

First,  it  lowers  the  freezing  point  and  by  so  doing  gives  the  con- 
crete a  better  opportunity  to  attain  its  initial  set  before  freezing. 

Second,  it  tends  to  retard  the  setting  and  allows  the  materials, 
the  cement,  water  and  aggregate  to  be  heated  to  a  somewhat  higher 
temperature  than  would  be  permissible  were  it  not  for  the  salt  in 
the  mix. 

Third,  since  salt  has  an  affinity  for  water,  it  retains  in  the  concrete 
the  necessary  moisture  required  for  perfect  crystallization.  In 
other  words,  it  prevents  the  concrete  from  drying  out  before  it  has 
had  time  to  set  when  it  thaws  after  freezing. 

Calcium  chloride  has  also  been  used  to  some  extent  to  prevent 
the  freezing  of  concrete  in  cold  weather,  but  owing  to  the  fact  that 
common  salt  is  so  much  less  expensive  and  more  readily  obtained, 
it  is  almost  universally  used  by  those  accustomed  to  do  work 
in  the  winter  season. 

The  use  of  salt  in  the  concrete  does  not  appear  to  impair  its 
strength  in  the  least,  nor  does  it  appear  to  have  an  injurious  effect 
on  the  metal,  provided  the  metal  is  well  covered  with  wet  concrete. 

The  use  of  brine  in  the  mixture  is  particularly  advantageous  in 
placing  mass  concrete  in  preventing  scaling  of  the  surface  from  frost 
action  and  while  it  may  somewhat  retard  the  setting  and  hardening 
the  ultimate  result  appears  to  be  a  concrete  of  even  greater  strength 
than  that  hardened  under  nominally  more  favorable  conditions  of 
warm  weather. 

Concrete  in  setting  generates  considerable  heat  after  the  action 
of  setting  has  started.  In  cold  weather  it  requires  artificial  heating 
to  start  this  chemical  action.  An  experiment  was  tried  on  one 
piece  of  work  when  the  temperature  outside  was  about  25  below  zero. 
A  piece  of  gas  pipe  was  inserted  into  a  column  36"  square,  just  cast. 
A  thermometer  was  dropped  down  the  pipe  three  feet  and  the  upper 
end  sealed  with  a  cork.  Upon  removing  the  thermometer  twelve 
hours  afterwards  it  registered  95  degrees  Fahr. 

16.     Curing  Concrete  Where  Proper  Precautions  Have  Not  Been 

Taken 

The  engineer  is  frequently  called  upon  to  pass  upon  concrete  which 
has  been  placed  and  the  precautions  heretofore  recommended  have 
not  been  followed. 


PRECAUTIONS    IN    SPLICING  31 

We  have  known  of  cases  where  the  concrete  was  placed  in  Decem- 
ber, mixed  with  cold  water,  frozen  as  fast  as  placed,  and  when  this 
same  material  thawed  out  in  March  it  was  as  soft  as  the  day  when 
first  cast.  After  the  concrete  had  been  kept  thoroly  wet  for 
two  and  one-half  weeks  and  then  allowed  to  dry  out,  a  good  hard 
concrete  was  secured  which  after  eight  months  stood  an  exceptionally 
satisfactory  test. 

Concrete  unless  promptly  thawed  out  after  it  has  been  frozen, 
sets  so  slowly  that  its  hardening  may  be  condemned  as  altogether 
too  slow  for  practical  purposes  if  it  is  expected  to  clean  up  the  work 
and  get  it  finished  within  a  reasonable  time,  and  for  this  reason 
it  pays  the  constructor  well  to  heat  the  materials  so  that  the  centers 
may  be  removed  promptly  and  the  work  finished  up  nearly,  if  not 
quite,  as  rapidly  as  it  is  ordinarily  done  in  the  summer. 

Too  great  care,  however,  cannot  be  exercised  in  handling  work 
during  the  winter  season  since  frozen  or  partly  frozen  concrete  may 
stand  well  when  the  forms  are  first  removed  and  then  as  soon  as  it 
commences  to  thaw  it  will  begin  to  get  out  of  shape  and  look  badly 
if  it  does  not  entirely  collapse. 

Many  mistakes  in  judgment  are  made  in  handling  work  in  the 
cold  season  of  the  year,  although  by  the  exercise  of  care  and  good 
judgment  there  is  no  reason  why  the  work  cannot  be  executed  in 
a  thoroly  first  class  and  satisfactory  manner. 

In  working  during  the  winter  season  snow  and  ice  frequently  get 
on  the  forms.  This  can  be  readily  removed  by  the  use  of  a  steam 
hose,  melting  the  snow  and  ice  in  advance  of  placing  the  concrete. 

17.     Precautions   in   Splicing,   Mixing,   Heating,    Etc. 

Attention  is  called  pointedly  to  the  necessity  of  melting  snow  and 
ice  on  old  work  and  on  forms  before  casting  concrete  and  it  remains 
to  call  attention  to  the  necessity  of  special  care  in  the  splicing  work. 

The  old  concrete  may  be  frozen  and  not  hardened.  It  will  be 
killed  or  disintegrated  by  heating  with  hot  water  as  some  thought- 
less foremen  have  tried  to  do.  Splices  in  the  work  should  be  made 
with  great  care  and  in  a  vertical  plane  both  for  beams  and  slabs. 
The  old  concrete  should  be  cleaned  of  snow  and  ice  with  a  steam  hose, 
but  no  hot  water  used,  then  the  new  concrete  may  be  cast  against 
it  and  the  moderate  temperature  of  the  new  concrete  will  gradually 
soften  the  old  work  if  frozen  and  the  result  will  be  a  satisfactory  bond 
between  the  two.  It  is  preferable  where  practicable  to  continue 
casting  until  a  whole  floor  is  complete  unless  the  work  is  of  too  great 
magnitude. 


32  REMOVAL    OF    FORMS 

Inclined  splices  and  irregular  joints  are  very  decided  sources  of 
weakness  in  work  cast  in  cold  weather,  in  fact,  they  can  hardly  be 
made  good  unless  by  digging  out  some  of  the  concrete  and  thoroly 
grouting  the  joint  after  the  work  has  hardened. 

The  foreman  should  be  cautioned  against  killing  cement  by 
mixing  with  boiling  hot  water.  Mixing  the  sand  and  stone  first 
with  boiling  water  will  take  the  frost  out  of  the  stone  and  sand  and 
warm  it  up  and  reduce  the  temperature  of  the  mix  down  to  120  or 
130  degrees  which  will  not  injure  the  cement.  If  there  is  ample 
salt  this  temperature  may  be  even  ten  or  fifteen  degrees  higher 
for  a  few  minutes  and  not  materially  damage  the  mix. 

18.     Caution  Regarding  Removal  of  Forms. 

A  word  of  caution  to  the  builder  may  not  be  amiss  under  this 
heading.  The  rapidity  of  the  setting  of  concrete  and  hence  the 
time  at  which  it  is  safe  to  remove  the  forms  varies  materially,  depend- 
ing on  the  humidity  of  the  atmosphere.  In  damp,  rainy,  wet, 
chilly  weather,  concrete  is  liable  to  set  very  slowly  indeed.  In  dry 
weather,  and  particularly  in  high  altitudes  the  concrete  sets  much 
more  rapidly.  In  central  Minnesota  for  example,  under  usual 
conditions,  concrete  may  be  counted  upon  to  set  more  rapidly  at 
temperatures  ten  or  fifteen  degrees  lower  than  in  situations  close 
to  the  Great  Lakes  or  in  the  South  where  there  is  a  large  difference 
in  humidity.  The  experienced  superintendent  soon  becomes  familiar 
with  these  conditions  for  a  given  locality,  but  if  he  moves  about  it 
is  well  to  bear  these  general  facts  in  mind  since  he  will  find  a  marked 
difference  in  different  sections  even  with  the  same  cement. 

As  to  the  time  of  removal  of  the  forms,  the  builder  should  bear 
the  fact  clearly  in  mind  that  it  is  not  the  number  of  days  time  that 
the  concrete  has  been  in  place  or  has  stood  upon  the  forms  that 
determines  whether  it  is  safe  to  remove  the  forms,  but  the  degree 
of  hardness  that  has  been  attained  during  that  period.  Concrete 
may  remain  on  the  forms  for  four  months  in  a  northern  climate, 
freeze  and  thaw  out  in  spring  and  be  as  soft  as  the  day  on  which  it 
was  placed,  if  the  foreman  has  been  so  lacking  in  judgment  as  to 
use  cold  water  to  mix  the  concrete  and  then  allow  the  material 
to  freeze  after  it  has  been  placed. 

Frequently  as  far  south  as  southern  Kansas,  damp,  chilly  weather 
so  retards  the  setting  of  cement  not  mixed  with  warm  water  that 
after  the  forms  have  remained  in  place  a  month  the  construction  will 
not  hold  its  shape,  but  will  sag  materially,  owing  to  its  half-hard- 


AVOIDANCE    OF    ACCIDENT  33 

ened  condition.  This  will  never  occur  where  the  simple,  inexpensive 
precaution  has  been  used  of  warming  up  the  water  at  all  times  when 
the  temperature  is  below  45  degrees  Fahr. 

It  should  be  borne  in  mind  by  the  builder,  that  the  slightest  sag 
in  the  construction,  while  it  may  not  affect  the  strength  in  the  least, 
usually  causes  the  owner  to  be  suspicious  of  the  integrity  of  the 
whole  work,  and  as  the  constructor  depends  upon  satisfied  customers 
in  a  large  measure  for  future  business,  and  for  the  prompt  payment 
of  the  contract  price  for  the  work,  these  matters  should  receive  careful 
attention. 

The  danger  of  accident  with  half-hardened  concrete  is  compara- 
tively remote  with  multiple  way  systems,  as  this  type  of  construction 
will  almost  invariably  give  the  workman  ample  time  to  note  its 
yielding  and  to  prop  it  up  before  excessive  deflection  has  occurred. 
Unfortunately,  this  is  not  the  case  with  one-way  reinforcement. 
When  it  once  starts  yielding  as  a  rule  the  whole  construction  goes  by 
the  run,  and  hence  from  the  stand  point  of  safety  to  workmen,  the 
superintendent  should  exercise  extreme  care  with  this  type  of  con- 
struction. 

The  question  of  determining  the  hardness  of  the  concrete  and 
whether  it  is  safe  to  remove  the  forms  is  one  which  the  builder 
must  decide.  As  a  rough  test  the  concrete  should  be  so  hard  that 
a  twenty  penny  nail  will  double  over  and  cannot  be  driven  into 
it  more  than  three-quarters  of  an  inch.  A  good  idea  can  be  obtained 
as  to  its  hardness  by  trying  it  with  a  hammer  and  seeing  how  readily 
it  can  be  indented,  as  well  as  by  driving  a  nail  into  it  and  finding 
out  its  condition  under  the  surface.  Examining  the  concrete  around 
openings,  etc.,  will  enable  the  experienced  foreman  to  form  a  correct 
judgment  as  to  whether  it  is  safe  to  remove  the  centering.  In  any 
case,  these  rough  tests  are  sufficient  to  determine  whether  the  removal 
of  the  forms  is  safe  for  the  workmen. 

In  long  span  slabs  or  beams  there  may  be  some  sag  owing  to 
compression  of  the  concrete  if  it  has  not  set  sufficiently  hard  altho 
no  accident  may  result.  Such  deflection  or  sagging  tends  to  destroy 
the  owner's  confidence  in  the  work  altho  it  may  have  no  material 
effect  from  the  standpoint  of  strength.  In  fact,  where  a  long  span 
slab  or  beam  has  sagged  a  moderate  amount  before  the  concrete  is 
thoroly  hard  there  is  generally  little  loss  of  strength. 

In  the  case  of  a  slab,  if  it  is  afterwards  leveled  up  with  additional 
concrete,  it  is  stronger  than  that  portion  of  the  work  which  has 
kept  its  shape.  The  owner  considers  this  an  evidence  of  weakness. 


34  REINFORCING    STEEL 

The  builder  knows  that  if  he  has  filled  up  the  depression  in  a  panel 
which  has  sagged  slightly  it  is  probably  the  strongest  slab  in  the 
building  and  a  test  of  it  will  give  exceptionally  fine  results. 

The  time  during  which  the  centering  should  remain  in  place  varies 
for  different  spans.  With  a  slab  sixteen  or  seventeen  feet  square  and 
seven  inches  thick  under  favorable  drying  conditions,  it  should  be 
possible  to  remove  the  forms  in  eight  or  ten  days.  Where  the  span 
is  longer,  say  twenty  or  twenty-five  feet,  two  to  three  weeks  at  least 
should  be  allowed  unless  the  slab  is  extra  thick.  For  example,  a 
slab  eight  inches  in  thickness  and  twenty-five  feet  in  span  should  be 
allowed  three  weeks  under  the  most  favorable  conditions  to 
thoroly  harden  if  it  is  to  keep  its  shape  immediately  after  removal 
of  the  supporting  forms.  Whereas  a  span  of  the  same  length, 
thirteen  inches  thick  would  only  need  practically  the  same  time  as 
the  shorter  span  on  account  of  the  additional  thickness.  These  are 
practical  points  which  it  is  well  to  bear  in  mind  as  upon  them  commer- 
cial success  in  a  measure  depends. 

The  superintendent  should  bear  in  mind,  in  freezing  weather, 
that  concrete  is  as  readily  stiffened  up  by  frost  as  by  the  true  chemical 
action  of  hardening  and  that  when  thus  hardened  by  frost  it  only 
remains  for  a  rise  in  temperature  to  occur  for  the  work  to  get  out  of 
shape,  if  it  does  not  actually  collapse.  The  test  in  freezing  weather 
to  determine  whether  the  concrete  has  been  merely  stiffened  up  b}' 
cold  or  is  actually  cured,  is  to  dig  out  a  small  sample  and  place  it  upon 
a  hot  stove,  then  if  it  sweats  and  softens  the  forms  must  remain  in 
place.  -If,  on  the  other  hand,  it  remains  hard  and  rigid  and  does 
not  sweat  and  soften  up  then  the  concrete  may  be  depended  upon 
to  retain  its  shape  and  forms  may  be  safely  removed. 

19.     Reinforcing  Steel 

Steel  for  reinforcement  should  be  tough,  homogeneous  metal, 
preferably  of  structural  steel  grade,  where  bending  is  required,  or 
of  harder  grade  for  slab  rods  when  bending  is  unnecessary. 

The  following  are  the  specifications  adopted  by  the  Association  of 
American  Steel  Manufacturers,  1910,  governing  the  chemical  and 
physical  properties  of  concrete  reinforcing  bars: 

Standard    Specifications   for   Concrete    Reinforcement    Bars 

1.  Manufacture.     Steel  may  be  made  by  either  the  open-hearth 
or  Bessemer  process.     Bars  shall  be  rolled  from  billets. 

2.  Chemical  and  Physical  Properties.     The  chemical  and  physical 
properties  shall  conform  to  the  following  limits : 


STEEL   SPECIFICATIONS 


35 


Properties 
Considered 

Structural  Steel  Grade 

Hard  Grade 

Cold- 
Twisted 
Bars 

Plain 
Bars 

Deformed 
Bars 

Plain 
Bars 

Deformed 
Bars 

Phosphorus,  maximum, 
Bessemer  
Open-hearth  

Ultimate    tensile 
strength,     pounds 
per  sq.  in 

.10 
.06 

55-70,000 

33,000 
1,400,000 

.10 
.06 

'  ,    ! 

55-70,000 

33,000 
1,250,000 

.10 
.06 

80,000  min. 

50,000 
1,200,000 

.10 
.06 

80,000  min. 

50,000 
1,000,000 

.10 
.06 

Recorded 
only 

55,000 

5% 

180°d.=2t. 
180°d  =3t. 

Yield  point,  minimum 
pounds  per  sq.  in  . 

Elongation,   per  cent 
in  8",  minimum  .  .  . 

Cold     bend    without 
fracture  : 
Bars   under    I"   in 
diameter  or  thick- 
ness 

T.  S. 

180°d.=lt. 
180°d.=lt. 

T.  S. 

180°d.=lt. 
180°d  =2t. 

T.  S. 

180°d.=3t. 
90  °d  =3t. 

T.  S. 

180°d.=4t. 
90  °d  =4t. 

Bars  \"  in    diameter 
or  thickness  and 
over.  . 

The  hard  grade  will  be  used  only  when  specified. 

3.  Chemical    Determinations.     In    order    to    determine    if    the 
material  conforms  to  the  chemical  limitatioDs  prescribed  in  para- 
graph 2,  herein,  analysis  shall  be  made  by  the  manufacturer  from 
a  test  ingot  taken  at  the  time  of  the  pouring  of  each  melt  or  blow  of 
steel,  and  a  correct  copy  of  such  analysis  shall  be  furnished  to  the 
engineer  or  his  inspector. 

4.  Yield  Point.     For  the  purposes  of  these  specifications,  the 
yield  point  shall  be  determined  by  careful  observation  of  the  drop 
of  the  beam  of  the  testing  machine,  or  by  other  equally  accurate 
method. 

5.  Forms  of  Specimens,     (a)     Tensile  and  bending  test  speci- 
mens may  be  cut  from  bars  as  rolled,  but  tensile  and  bending  test 
specimens  of  deformed  bars  may  be  planed  or  turned  for  a  length  of 
at  least  9  inches  if  deemed  necessary  by  the  manufacturer  in  order 
to  obtain  uniform  cross-section. 

(6)  Tensile  and  bending  test  specimens  of  cold-twisted  bars 
shall  be  cut  from  the  bars  after  twisting,  and  shall  be  tested  in  full 
size  without  further  treatment,  unless  otherwise  specified  as  in  (c), 
in  which  case  the  conditions  therein  stipulated  shall  govern. 


36  STEEL    SPECIFICATIONS 

(c)  If  it  is  desired  that  the  testing  and  acceptance  for  cold- 
twisted  bars  be  made  upon  the  hot  rolled  bars  before  being  twisted, 
the  hot  rolled  bars  shall  meet  the  requirements  of  the  structural 
steel  grade  for  plain  bars  shown  in  this  specification. 

6.  Number  of  Tests.     At  least  one  tensile  and  one  bending  test 
shall  be  made  from  each  melt  of  open-hearth  steel  rolled,  and  from 
each  blow  or  lot  of  ten  tons  of  Bessemer  steel  rolled.     In  case  bars 
differing  f  inch  and  more  in  diameter  or  thickness  are  rolled  from 
one  melt  or  blow,  a  test  shall  be  made  from  the  thickest  and  thinnest 
material   rolled.     Should   either   of   these   test    specimens    develop 
flaws,  or  should  the  tensile  test  specimen  break  outside  of  the  middle 
third  of  its  gauged  length,  it  may  be  discarded  and  another  test 
specimen  substituted  therefor.     In  case  a  tensile  test  specimen  does 
not  meet  the  specifications,  an  additional  test  may  be  made. 

(d)  The  bending  test  may  be  made  by  pressure  or  by  light  blows. 

7.  Modifications  in  Elongation  for  Thin  and  Thick  Material.     For 
bars  less  than  7/16  inch  and  more  than  f  inch  nominal  diameter 
or  thickness,  the  following  modifications  shall  be  made  in  the  require- 
ments for  elongation: 

(e)  For  each  increase  of  f  inch  in  diameter  or  thickness  above 
f  inch,  a  deduction  of  1  shall  be  made  from  the  specified  percentage 
of  elongation. 

(/)  For  'each  decrease  of  1/16  inch  in  diameter  or  thickness  below 
7/16  inch,  a  deduction  of  1  shall  be  made  from  the  specified  percentage 
of  elongation. 

(</)  The  above  modifications  in  elongation  shall  not  apply  to 
cold-twisted  bars. 

8.  Number  of  Tests.     Cold-twisted  bars  shall  be  twisted  cold 
with  one  complete  twist  in  a  length  equal  to  not  more  than  12  times 
the  thickness  of  the  bar. 

9.  Finish.     Material  must  be  free  from  injurious  seams,  flaws, 
or  cracks,  and  have  a  workmanlike  finish. 

10.  Variation  in  Weight.     Bars  for  reinforcement  are  subject 
to  rejection  if  the  actual  weight  of  any  lot  varies  more  than  5  percent 
over  or  under  the  theoretical  weight  of  that  lot. 

20.     Quality  of  Steel 

Unfortunately  there  is  an  idea  prevalent  that  almost  any  grade  of 
metal  is  good  enough  for  reinforcement.  Where  the  contractor  or 


QUALITY    OF    STEEL  |J7 

engineer  is  responsible  for  the  test  strength  and  permanence  of  the 
work  he  needs  to  see  that  the  steel  is  of  suitable  quality. 

The  product  of  what  is  called  a  fagot  mill  is  generally  very  un- 
desirable. The  trade  term,  sometimes  applied  to  this  product  is 
" Bushel  Steel."  A  fagot  is  formed  using  muck  bar  iron  flats  for 
bottom  and  sides  and  filling  in  with  miscellaneous  scrap  steel,  iron, 
etc.,  heating  up  and  rolling  into  billets  and  bars.  This  grade  of 
metal  has  an  ultimate  strength  of  about  45,000  pounds  per  square 
inch  and  a  commercial  yield  point  of  25,000  pounds.  It  will  bend 
easily  but  when  nicked  and  broken  will  show  a  dull  fracture  of  a 
ragged  and  coarse  texture  in  strong  contrast  with  the  bright,  fine 
crystalline  or  silky  texture  of  a  good  grade  of  steel. 

Rerolled  rail  stock  is  sold  to  a  large  extent  for  reinforcing  metal. 
Small  rods  rolled  from  the  flange  and  stem  of  the  rails  make  excellent 
slab  reinforcement  while  those  which  are  rolled  from  the  heads  are 
liable  to  be  brittle  and  unreliable.  This  grade  of  steel  runs  from 
80,000  to  125,000  pounds  per  square  inch  ultimate  strength  and  is 
much  too  hard  to  be  safely  bent  cold. 

Cold  twisting  is  safe  only  with  a  soft  or  medium  soft  grade  of 
metal.  The  effect  of  cold  twisting  is  to  raise  the  yield  point,  reduce 
toughness  and  elongation  thus  rendering  the  metal  more  brittle  and 
unreliable  and  for  that  reason  objectionable.  Where  the  mechanical 
bond  of  a  twisted  bar  is  demanded  hot  twisting  is  preferable  for  the 
above  reasons. 

Hard  grade  steel  has  a  decided  advantage  for  slab  rods  of  small 
diameter,  since  the  harder  the  metal  the  less  liable  they  are  to  kink 
in  the  handling  and  shipping.  Specifications  for  small  bars  such  as 
5/ 16  to  J  inch,  should  require  that  these  bars  be  shipped  in  bundles 
of  a  dozen  to  fifteen  rods  per  bundle  well  wired  together,  so  that 
they  will  be  less  liable  to  be  bent  in  shipment  and  can  be  more 
readily  handled  on  the  work. 

21.     Cold  Bending  With  Mild  Steel 

In  bending  rods  for  beams,  columns  or  slabs,  the  method  used 
depends  somewhat  on  the  character  of  the  steel.  With  the  kind 
of  metal  reinforcement  recommended,  namely,  medium  steel,  nearly 
all  of  the  work  of  bending  is  done  cold  and  at  a  comparatively  in- 
significant cost.  For  instance,  bending  the  column  rods  for  the 
Mushroom  system  on  one  of  the  large  pieces  of  work  cost  about 
fifty  cents  per  ton.  In  this  case  a  bending  machine  was  arranged 


38 


BENDING    STEEL 


using  gears  similar  to  those  of  the  ordinary  crab  hoist,  bending  the 
bars  by  means  of  a  crank  pin  on  the  driven  shaft.  Bars  are  not 
damaged  to  any  considerable  extent  provided  that  the  radius  of 
the  bend  is  not  too  sharp  and  that  the  moving  part  bending  the  bar 
does  not  jam  the  metal  so  that  its  flow  is  confined  to  a  short  section. 

This  is  one  of  the  difficulties  with  quite  a  number  of  the  lever 
bending  machines  which  we  have  investigated.  Bars  were  found  at 
one  building  which  were  cracked  at  the  bend.  Knowing  the  metal 
to  be  good,  tough,  medium  steel,  the  bending  machine  was  imme- 
diately investigated.  It  was  found  that  a  die  with  a  sharp  corner  had 
been  used  around  which  to  bend  the  bar.  One  or  two  bars  were 
broken  in  handling  after  bending.  Fortunately  none  of  them  were 
in  a  position  where  direct  tensile  stress  came  upon  the  metal. 
This  die  was  immediately  ordered  to  be  cut  over  to  a  reasonable 
radius  of  one  and  one  half  diameters  of  the  bars. 

22.     Bending  Machines 

Light  rods,  such  as  f  inch  and  J  inch  and  the  like,  may  be  readily 
bent  by  the  use  of  tongs  or  a  short  piece  of  pipe  slipped  over  the  end 
of  the  rod.  Such  tongs  are  illustrated  in  the  accompanying  Fig.  4, 
and  also  a  lever  bender  for  long  rods. 


Fig.  4.     Cut  of  Lever  Bender  and  Tongs. 

In  general,  a  good  detail  for  a  lever  bender  is  to  arrange  a  small 
roller  on  the  moving  part  of  the  bending  lever  so  that  the  pressure 
is  brought  against  the  bar  by  a  roll  and  so  that  there  may  be  no 
tendency  to  localize  the  stretch  of  the  metal  at  one  place  by  friction, 
thereby  seriously  injuring  the  bar. 

For  ring  rods  and  the  like  such  as  are  used  in  the  mushroom 
system,  an  ordinary  set  of  blacksmith's  tire  rolls  is  the  most  con- 
venient equipment. 

For  spirals  the  same  set  of  rolls  is  frequently  used. 


KKNDING    MACHINES 


39 


Fig.  5-a. 
Hand  Power,  Star  Bender,  Bending  Column  Rods. 


The  accompanying  three  figures  show  a  very  convenient  bending 
machine,  manufactured  by  Kardong  Bros.,  of  Minneapolis,  Minn. 

Fig.  5a  shows  the  bending  of  a  mushroom  column  rod,  the  stop 
on  the  circular  segment  fixing  the  angle  of  the  bend  in  a  hand 
power  machine. 

Fig.  5b  shows  a  form  of  the  machine  arranged  with  a  gas  engine 
power  for  rapid  operation. 

Fig.  5c  is  a  view  showing  the  bending  of  beam  rods  and  spiral 
hooping  with  this  machine. 


Fig.  5b. 
Gasoline  Power,  Star  Bender,  Bending  Beam  Rods  and  Spirals. 


10 


MOT    IIKNDINd 


.       . 
Showing  I '..-I  i  Side  of  St:ir  Bender. 

23.     Hot  (tending  and  Precautions  with  High  Carbon  Steel 

In  bending  burs  hoi,  which  is  done  commonly  where  the  burs 
are  luird  steel  over  1  inch  diameter,  there  is  sometimes  carelessness 
in  overheating  the  steel.  Heating  up  to  a  low  cherry  red  is  the 
highest  which  should  he  permitted  by  the  foreman  in  charge  of  such 
work. 

High  carbon  steel  is  more  readily  injured  by  overheating  than 
mild  steel.  It  is  too  hard  to  be  worked  cold  and  can  only  be  bent 
to  the  desired  form  by  heating.  In  heating  it  is  much  more  liable 
to  be  severely  damaged  than  mild  steel  and  hence  extra  care  should 
be  taken  when  using  this  grade  of  metal,  to  see  that  it  is  not  burned 
by  the  smith. 

In  bending  cold  twisted  bars,  where  specified  by  the  architect, 
they  should  be  invariably  heated,  otherwise  in  endeavoring  to  bend 
them  cold  the  damage4  done  to  the  bar  by  the  cold  twisting  will  mani- 
fest itself  in  brittleness  of  the  bar  and  the  tendency  to  crack  or  break 
at  the.  point  where  the  bend  is  being  made. 

24.    Centering 

Centering,  or  false  work,  is  one  of  the  largest  elements  of  cost 
in  reinforced  concrete  construction,  and  accordingly  one  that  should 
receive  careful  consideration  from  the  standpoint  of  cost,  and  also 
from  the  standpoint  of  the  safety  both  of  the  work  and  of  the  men 
putting  up  the  work. 

We  will  first  consider  centering  for  flat  slab  and  column  con- 
struction. Seven-eights  inch  lagging  with  2x(>"  or  2x8"  joists  requires 


CKNTKRING  41 

the  least  lumber  and  is  cheaper  with  lumber  prices  from  $18  to  $22 
per  thousand  such  as  rule  in  the  middle  and  Kastern  states  at  present, 
but  with  lumber  at  from  $8  to  $12  per  thousand,  plank  from  ledger 
ledger  would  be  more  economical,  the  latter  price  being  common 
at  such  Pacific  Coast  points  as  Vancouver,  Seattle,  etc;. 

Wide  boards  are  undesirable  for  lagging,  since  they  warp  in 
the  sun  and  swell  excessively  so  that  they  get  out  of  shape,  leaving 
an  uneven  surface;  a  lx(>"  lagging  of  No.  1  common  fencing  SI  S2K  is 
preferable  for  the;  foregoing  reason.  Matched  boards  should  not  be 
used  as  the  grooves  are  readily  broken  out  and  leave  a  rough  surface. 

Joists  for  an  8"  slab  should  be  not  less  than  2x0",  si/ed  and  about 
22"  centers  for  spans  of  0  feet  from  ledger  to  ledger,  and  2x8"  sized 
for  spans  of  7  feet  to  8  feet  center  to  center. 

Spacing  of  ledgers  should  be  arranged  with  reference  to  the 
column  spacing,  so  that  the  line  of  columns  will  come  approximately 
at  the  center  of  a  span  between  ledgers.  Thus  for  columns  20  feel 
centers,  (the  joists  running  in  UK;  direction  of  the  20  foot  span)  a 
spacing  of  (>'  8"  for  ledgers  is  economical,  using  2x8"  joists,  while 
if  the  span  is  18  feet,  2x(>"  joists  with  (i  foot  spacing  of  ledgers  is  best. 

Working  stress  for  pirn;  and  common  lumber  should  not  exceed 
800  pounds  per  square  inch  fiber  stress,  and  due  regard  must  be  had 
for  the  stiffness  of  the  work,  while  for  Douglas  fir  and  long  leaf 
yellow  pine  forty  percent  higher  stresses  are  permissible. 


Led(jc/rx  and  J'oNl 

Evidently  the  fewer  the  points  of  support  the,  less  will  be  the 
work  of  leveling  up  the  forms.  Four  by  four  posts  based  on  wedges 
12  inches  long,  cut  by  splitting  a  4x4",  three  inches  at  the  thick  end 
by  one  inch  at  the  thin  end  is  the  simplest  practical  adjustment. 

For  economy  the  ledgers  should  be  deep,  but  attention  must  be 
paid  to  the  tendency  of  a  narrow  but  deep  beam  to  cripple;  by  buck- 
ling of  the  compression  side;  or  top  face.  Many  centering  failures 
occur  from  this  cause,  where  a  2x10"  or  a  2x12"  has  been  selected  by 
a  table  of  loads  as  having  a  sufficient  capacity.  Then  the  supports 
an;  placed  six  or  eight  feet  between,  centers,  and  unstayed,  when,  if 
the  ledger  twists  the  failure  is  so  sudden  as  to  give  little  if  any  warning. 

The  unsupported  width  ought  not  to  exceed  thirty  times  the; 
thickness,  as  a  practical  dimension,  the  support  of  the  bearing  joist 
not  being  counted  unless  the  ledger  is  held  laterally  by  cleats  to  the 
joist. 


42  FORMULAS    FOR    DESIGN    OF    POSTS    AND    BEAMS 

When  a  deep  ledger  is  used,  as  two  2x10"  or  two  2x12",  it 
should  be  double;  the  verticals  should  be  stayed  at  the  underside 
of  the  ledger  with  a  light  strip  of  1x4"  running  transversely  to  the 
ledger  from  post  to  post  and  continuous  through  the  extent  of  the 
centering. 

Formulas  for  Proportioning  Beams  and  Posts 
A  convenient  formula  for  the  capacity  of  joists  and  ledgers  as 
simple  beams  is,  for  pine  or  hemlock. 

W  =100  b  d2  /L,  in  which 

W  =  capacity  of  the  joist  or  ledger  in  pounds  for  uniform 
loading. 

b  =  breadth  in  inches. 

d  =  depth  in  inches 

L  =  span  in  feet. 

When  full  continuity  is  secured  over  two  spans,  twenty-five  percent 
can  be  safely  added  to  this  capacity. 

The  above  formula  is  applicable  to  plank  flatwise  as  far  as  the 
fiber  stress  is  concerned,  but  the  same  fiber  stress  for  a  plank  or 
board  flatwise  will  give  too  great  a  deflection,  so  that  the  plank  or 
board  must  be  figured  or  selected  for  stiffness  in  keeping  with  span. 

For  Douglas  fir  or  yellow  pine,  fifty  percent  increase  in  the  above 
safe  load  is  permissible. 

Douglas  fir  or  yellow  pine  timber  of  4x4"  vertical  posts  may 
be  figured  as  safe  for  800  Ibs.  per  sq.  in.,  and  Norway  pine  or. 
spruce  for  600,  if  stayed  laterally  in  each  direction  by  stays  six  feet 
apart  center  to  center  vertically. 

A  convenient  formula  for  fir  or  yellow  pine  posts  is 
P=1000-10L/£ 

in  which  P  =  safe  load  in  Ibs.  per  square  inch,  L  =  unsupported  length 
between  lateral  stays  in  inches,  B  =  the  least  breadth  in  inches. 
For  Norway  pine  or  hemlock  take  six  tenths  of  the  above  values. 

Fig.  6  shows  the  centering  used  in  the  Minneapolis  Knitting 
Company  building,  a  structure  which  we  have  termed  type  III. 
The  joists  used  were  2x6s,  20  inch  centers,  with  1x6"  fencing  for 
the  floor.  For  studding  4x4s  are  usually  used,  spaced  about  seven 
feet  apart,  capped  by  2x8s  double  and  resting  on  wedges  by  means  of 
which  the  centering  can  readily  be  adjusted  to  the  desired  level  of 
the  finished  floor. 


CENTERING  43 

For  square  columns  of  small  section  2x4s  spiked  together,  forming 
the  square  ties,  are  about  as  cheap  as  any  method  of  putting  the 
boxes  together. 

For  columns  some  use  4x4"  side  pieces,  slotted  at  the  end  and  \" 
bolts.  This  allows  the  same  frame  to  be  adjusted  for  different  size 
columns  and  makes  a  very  substantial  form,  but  somewhat  expensive 
in  first  cost.  For  beam  boxes  If"  plank  for  bottom  and  J"  boards 
for  sides  are  preferable.  For  plain  slab  forms  the  following  is  the 
writer's  preference,  where  lumber  is  used: 

Joist  2x8",  20  to  22"  centers,  1x6"  fencing  for  sheathing, 
2xlOs  double  for  ledgers  spaced  eight  to  nine  feet  apart.  Vertical 


Fig.  G.      Centering  Northwestern  Knitting  Company  Building. 

posts  seven  to  eight  feet  centers.  The  4x4"  verticals  butted  under  the 
ledger  pieces  and  the  ledger  was  prevented  from  turning  on  top  of 
the  4x4s  by  short  pieces  of  Jx4",  nailed  to  both  ledger  and  top  of  the 
4x4s  with  8s"  nails.  The  bottom  of  the  posts  are  best  adjusted  by 
wedges  12"  long,  cut  from  4x4s.  This  will  allow  the  leveling  up  of 
the  centering  very  readily. 

In  centering  shown  in  Fig.  6  the  column  boxes  are  1J"  stock 
banded  by  2x4"  lapped  and  fastened  together  with  wire  spikes. 
Beam  boxes  were  made  up  of  f"  boards  and  2x4s  forming  vertical 
frame  and  1x6"  bottom  of  same.  A  light  ledger  is  nailed  along  the 
side  of  the  beam  box  to  receive  the  joist  for  supporting  the  slab.  The 


44  INSPECTION    OF    CENTERING 

beam  box  was  then  braced  up  and  two  lines  of  supports  placed  under 
the  2x6"  joist. 

Sometimes  it  is  desirable  to  center  by  using  sizes  of  lumber  which 
can  be  worked  into  boxing  such  as  is  used  for  hardware  storage  pur- 
poses, implement  house  requirements  and  the  like.  In  this  case 
verticals  can  readily  be  made  of  2x6s,  but  will  require  additional 
lateral  staying.  Verticals  are  usually  stayed  every  four  to  six  feet 
in  height  with  1x4"  ribbons  in  both  directions. 

Leveling  up  Centering 

Leveling  should  be  done  by  using  an  architect's  or  an  engineer's 
level. 

Evidently  the  fewer  verticals  there  are  the  more  readily  the  form 
can  be  leveled  up  and  placed  in  proper  condition  for  casting  con- 
crete. After  leveling  up,  the  wedges  should  be  nailed  so  that  there 
will  be  no  slipping.  The  vertical  studs  should  be  stayed  along  the 
line  of  the  joist  at  the  top  and  longitudinally  and  transversely  mid- 
way for  stories  ten  to  twelve  feet  in  height,  so  there  will  be  no  danger 
of  the  stud  kicking  or  buckling  and  the  centering  going  down  should 
a  heavy  car  run  off  the  track.  Where  the  area  to  be  centered  is  large 
it  sometimes  pays  to  cleat  the  sheathing  in  sections  two  feet  or  more 
wide.  This  eliminates  the  necessity  for  the  larger  part  of  the  nailing 
to  the  joist  and  enables  the  taking  down  of  the  forms  a  little  more 
readily. 

Wide  boards  should  not  be  used  for  sheathing  for  the  reason  that 
they  curl  and  split  badly  in  the  sun  and  swell  excessively  when  wet. 
For  that  reason  1x6"  square  edge  fencing  is  best.  Yellow  pine 
and  wood  which  will  stand  considerable  hard  usage  is  preferable 
to  hemlock  or  the  softer  grades  of  white  pine. 

Wetting  down  Wood  Centering 

Where  wood  sheathing  is  used  for  the  forms  it  should  be  thoroly 
wet  down  from  one  to  two  hours  in  advance  of  placing  the  concrete 
to  give  the  timber  which  has  probably  dried  out  in  the  sun,  a  chance 
to  swell  and  close  the  cracks  so  that  there  will  be  the  least  possible 
loss  of  cement  grout  as  the  casting  proceeds. 

Inspection  of  Centering  before  Casting  Concrete 
As  a  general  rule  the  foreman  should  be  instructed  to  inspect 
carefully  all  centering  before  starting  to  pour  the  concrete  for  the 
reason  that  many  of  the  stays  and  sometimes  some  of  the  verticals 
are  left  out  temporarily  for  convenience  in  erection,  with  the  ex- 
pectation of  putting  them  in  before  pouring  concrete  commences. 


COLUMN    FORMS 


Column  Forms 


45 


For  octagonal  forms  we  have  adopted  the  standard  shown  in 
the  Fig.  8  with  a  cast  iron  or  adjustable  sheet  metal  form  for  the  head. 


Fig.  8.     Column  Forms. 


The  column  box  is  bound  together  by  \"  rods  bent  in  semi- 
circular form,  with  a  long  thread  and  nut  at  the  end.  These  are 
passed  through  standard  malleable  clamps  used  for  wood  stove  pipe 
and  screwed  up. 

Another  method  of  making  up  column  forms  is  to  use  sheet  metal 
forming  adjustable  round  or  octagonal  heads,  see  Fig.  9. 

This  is  one  of  the  most  economical  types  of  column  forms.  It  is 
readily  handled,  weighs  but  little  and  costs  but  little  to  transport 
and  is  reasonable  in  first  cost. 

In  general,  a  light  sheet  metal  form  consists  of  sections  which 
are  adjustable  by  being  lapped  and  are  held  rigidly  by  heavy 
bands  of  quarter-inch  metal  at  intervals  of  about  two  feet. 

Sheet  Metal  for  Slab  Forms 

To  save  the  cost  of  sheathing  and  facilitate  rapid  handling  a 
large  amount  of  corrugated  steel  in  place  of  fencing  has  been  used 
for  decking. 


46  SHEET    METAL    FORMS 

Ceiling  of  this  type  is  shown  in  Fig.  10.  This  type  of  floor 
centering  is  not  suitable  where  it  is  desired  to  plaster,  but  for  a 
wholesale  building  or  in  fact  any  kind  where  special  decorative 
finish  is  not  desired  it  is  substantial  and  neat. 

Cost  of  handling  sheet  metal  is  about  one-third  that  of  laying 
boarding.     Greasing  it  with  paraffme 
oil   prevents   the   concrete  from    ad- 
hering and  facilitates  ready  removal 
and  rehandling. 

Advantages    that  are  claimed  for  sheet 
metal  forms  are  as  follows: 

That  the  sheet  metal  holds  the  moisture 
and  prevents  the  concrete  from  drying  out 
too  rapidly.  It  prevents  loss  by  leakage 
of  the  liquid  cement  mortar,  such  as  some- 
times occurs  where  board  forms  are  used, 
and  leaves  a  clean,  smooth  job. 

The  sheet  metal  centering  can  be  used 
over  and  over  again  and  should  it  be  bat- 
tered out  of  shape  it  is  a  comparatively 
inexpensive  matter  to  repress  the  sheets. 
At  first  cost  it  is  at  a  disadvantage  as  com- 
pared with  wood  centering,  but  in  the  long 
run  it  is  much  cheaper  for  the  reason  that  it 
is  lighter,  requires  less  labor  to  handle  and 
involves  less  labor  in  carting  from  point  to 
point.  The  guage  of  metal  should  not  be 
lighter  than  No.  20. 

Improper  Specifications  for  Centering 

Many  architects  have  a  totally  erroneous 
idea  as  to  the  proper  requirements  for  cen- 
tering.    For    example,    the   architects    fre- 
quently specify  matched  and  surfaced  lum-  Fig  g 
ber  for  forms,  with  the  vague  expectation  sheet  Metal  Column  Forms. 
that  by  so  doing  they  will  get  an  exception- 
ally smooth   job.      Unless  the  lumber  is  over  two  inches  thick, 
which  would   involve  an  unreasonably  great  expense,  the  tongue 
and   groove   will  be  soon  broken,  ragged  joints  and  edges  will  be  a 
frequent  rather  than  a  rare  occurrence,  and  on  the  whole  the  work 
will  not  present  so  smooth  an  appearance  as  though  ordinary  square 
edge  fencing  was  used  for  the  work. 


PARTIAL    REMOVAL    OF    FORMS  47 

If  it  is  required  that  the  work  be  finished  with  a  smooth  surface 
the  most  inexpensive  method  is  to  give  it  a  skin  coat  of  plaster  as 
recommended  in  the  sections  covering  the  subject  of  plastering  on 
reinforced  concrete. 

Partial  Removal  of  the  Forms 

It  is  evident  that  the  earlier  the  centering  can  be  removed  for 
use  in  the  upper  stories  the  less  material  will  be  necessary  in  handling 
the  work  and  the  lower  the  cost  if  successfully  executed.  In  the 
mushroom  system  it  is  customary  in  good  drying  weather  to  remove 
the  forms  in  from  twenty-four  to  forty-eight  hours  from  columns. 


Fig.  10.     View  showing  Ceiling  made  with  Corrugated  Steel  Forms, 
Con.  P.  Curran  Building,  St.  Louis,  Mo. 

In  this  type  of  construction  columns  carry  little  weight  until  after 
the  removal  of  the  slab  forms,  and  by  handling  the  work  in  this 
manner  a  very  much  smaller  number  of  forms  can  be  used  on  a 
large  job.  Where,  however,  beam  and  slab  forms  are  used 
the  column  generally  supports  the  beam  boxes  and  the  writer  is 
not  in  favor  of  removing  the  centering  in  part,  but  prefers  to  see 
the  whole  left  standing  together  except  perhaps  a  few  of  the  stays 
until  the  concrete  has  thoroly  cured. 

In  our  illustrations  of  rapidity  of  erection  of  reinforced  concrete 
a  number  of  examples  appear  which  indicate  clearly  the  number 


48  CENTERING CONTINUED 

of  floors  under  which  the  centering  is  left  in  the  conduct  of  work 
under  favorable  conditions. 

Handling  and  making  up  of  forms  is  more  a  question  of  craft 
than  of  figures.  As  to  the  question  of  ingenuity  the  brightest  en- 
gineer can  as  a  rule  learn  something  from  any  foreman  and  even  a 
good  carpenter  with  whom  he  comes  in  contact  in  this  line  of  work. 
Frequently,  however,  we  see  workmen  who  lack  ingenuity  and  a 
conception  of  the  simple  requirements  of  form  work.  For  example, 
we  occasionally  see  a  gang  of  carpenters  putting  up  an  expensive 
braced  form  for  a  thin  wall,  where  all  that  is  necessary  to  do  is  to  set 
up  the  cleated  boards  and  tie  them  together  with  No.  10  wire.  The 
pressure  on  the  two  sides  balance  and  the  need  of  bracing  is  practical- 
ly nil. 

Special  forms,  such  as  are  used  for  chimneys,  are  very  advantage- 
ously made  up  with  sheet  metal  and  arranged  to  be  slipped  upward 
as  the  work  advances.  It  is  hardly,  however,  within  the  scope  of 
this  work  to  go  into  special  constructions  of  this  character. 


49 


CHAPTER  II 
GENERAL  TYPES  OF  CONCRETE  FLOOR  CONSTRUCTION 

1 .  Classification — The  history  of  the  development  of  structural 
work  shows  that  the  engineer  has  been  largely  influenced  in  his 
first  efforts  to  design  any  new  type  by  the  forms  of  construction 


i 


Fig.  11.      Type  I. 


to  which  he  has  been  previously  accustomed.  For  example, 
when  wrought  iron  began  to  be  used  in  place  of  timber  for  railroad 
trestles  the  longitudinal  bracing  was  identical  with  that  used  in  timber 
construction;  indeed  it  was  at  first  gravely  questioned  whether 
these  braces  ought  not  to  be  made  of  timber  for  fear  of  the  unknown 
dangers  that  might  arise  from  the  unequal  expansion  of  these  braces 
of  iron  and  the  foundation  on  which  the  trestle  was  supported,  and 


50 


TYPES  OF  CONCRETE  FLOORS 


today  not  a  few  of  our  concrete  theorists  are  deeply  concerned 
regarding  equally  insignificant  questions. 

The  common  types  of  concrete  steel  floor  construction  may  be 
classified  as  follows: 

I.  The  earliest  type  of  timber  construction  has  been  followed 
or  imitated  closely  in  some  of  the  pioneer  structures  in  concrete 
steel  and  also  in  not  a  few  of  our  buildings  even  today.  This  type 
may  be  described  as  employing  columns  to  support  parallel  main 
girders  with  joists  in  one  direction  only  extending  crosswise  from 
girder  to  girder  and  a  thin  floor  covering  the  joists.  See  Fig.  11. 


Fig.  12.     Type  II. 


II.  Similar  to  I,  except  the  substitution  of  a  slab  without  joists 
from  girder  to  girder,   similar  to  mill  construction  of  beams  and 
thick  plank  flooring  from  beam  to  beam.     See  Fig.  12. 

III.  A  natural  concrete  type,  a  true  monolith,  departing  from 
the  characteristics  of  timber  and  steel  construction  in  the  employ- 
ment of  concrete  beams  from  column  to  column  in  two  directions 
and  slabs  with  panels  supported  on  four  sides.     See  Fig.  13. 

IV.  A  second  distinctively  concrete  type,  in  which  the  centering 
is  simplified  to  the  limit  and  consists  only  of  a  temporary  flooring  on 
which  to  pour  the  concrete.     The  elements  involved  are  two  only: 
column  supports,  and  a  continuous  flat  slab  supported  directly  by  the 
columns  and  integral  therewith.     See  Fig.  14. 


TYPES  OF  CONCRETE  FLOORS 


51 


A  modification  of  types  I  and  II  is  sometimes  employed  in  which 
arches  spring  from  girder  to  girder.  This  modification  is  not  a 
common  type  of  construction  however. 

As  to  safety,  these  types  must  be  rated  with  reference  to  their 
deportment  under  overload,  whether  failure  can  occur  suddenly  and 
without  warning,  or  slowly  and  gradually. 

Failure  is  more  rapid  where  the  flexure  under  load  has  a  single 
curvature  only  under  load  than  where  there  is  double  curvature. 
For  example,  a  slab  supported  on  two  sides  is  deformed  in  a  cylindrical 


Fig.  13.     Type  III. 


surface.  The  slab  supported  on  four  sides,  on  the  other  hand, 
dishes  or  bags  down  from  all  directions  and  cannot  fail  suddenly 
for  this  reason. 

Ample  lap  of  reinforcement  over  the  supports,  thoroly  tying 
the  work  together,  enhances  the  safety  of  all  types. 

Most  failures  entailing  loss  of  life  have  occurred  with  reinforce- 
ment in  one  direction  only  and  of  these  failures  the  greater  part  of 
them  have  occurred  where  insufficient  lap  of  reinforcement  has  been 
provided. 


52 


PROBLEMS    IN    DESIGN 


From  the  fireproof  standpoint,  that  form  which  exposes  the 
least  area  to  heat,  which  presents  the  most  uniform  distribution  of 
metal  to  provide  for  the  temperature  stresses  resulting  from  unequal 
heating  will  rank  first,  and  on  this  basis  the  natural  concrete  types 
III  and  IV  consequently  are  to  be  preferred. 

In  the  above  types,  we  have  the  following  five  problems  in  design : 

(a)  Beams,  simple,  continuous,  partially  continuous,  etc. 

(b)  Slab  with  panels  supported  on  two  sides. 


Fig.  14.      Type  IV. 


(c)  Slab  with  panels  supported  on  four  sides. 

(d)  Slab  with  panels  supported  on  four  posts  or  corners. 

In  each  of  the  slab  problems  we  also  must  consider  the  conditions 
of  the  ends  or  edges  of  the  panels  as  in  the  treatment  of  beams. 

(e)  Columns  similar  for  all  types. 

The  relative  economy  of  the  several  types  will  appear  from  the 
methods  of  computation  to  follow. 

2.  Utility  of  the  Theory  of  Action  of  Structures — No  theory  can 
be  devised  which  will  take  into  consideration  all  of  the  phenomena 
presented  by  an  actual  structure.  In  structural  work  it  is  usual  to 
treat  for  example,  beams  connected  by  flange  angles  or  resting  on 


PRINCIPLE    OF    PROPORTION  53 

top  of  girders  as  simple  beams.  The  stresses  in  such  beams,  however, 
differ  somewhat  from  what  they  would  be  on  the  theoretical  assump- 
tion that  the  supports  are  knife  edge  bearings  without  friction. 
The  useful  theory  then  is  that  which  takes  into  consideration  the 
predominant  phenomena  presented  by  the  structure  under  load. 
The  unnecessary  refinement  of  taking  into  account  small,  subsidiary 
actions,  such  as  the  restraint  of  connection  angles,  and  restraint  of 
beams  which  are  not  supported  by  frictionless  knife  edges,  is  for 
practical  purposes  ignored. 

Thus,  the  direct  tensile  resistance  of  the  concrete  as  a  tension 
chord  being  small,  is  disregarded  in  practical  computations.  In 
reinforced  concrete  beams  the  tensile  flange  resistance  offered  by 
the  steel  alone  as  a  flange  is  that  considered  and  counted  upon  for 
safety  as  the  predominant  action. 

The  theory  of  work  has  this  striking  advantage  over  other  methods 
of  analysis  of  such  structures,  that  it  indicates  this  predominant 
action  almost  at  once  in  a  manner  so  clear  that  it  requires  little  or 
no  computation  to  arrive  at  a  correct  method  of  treating  the  structure 
under  discussion. 

3.  Principle  of  Proportion — In  the  early    development    of    re- 
inforced concrete  work,   constructors  were  obliged  to  experiment  in 
order  to  ascertain  the  most  suitable  proportions  and  arrangement  of 
materials  for  supporting  a  given  load  on  a  given  span,  and  from  ex- 
periments of  this  kind  determine  by  proportion  the  carrying  strength 
for  other  loads  and  other  spans.     This  principle  of  proportion  is 
indeed  a  most  useful  one,  and  was  employed  largely  by  the  builders 
in  the  middle  ages  in  the  construction  of  masonry  work  in  the  form 
of  arches  in  the  great  cathedrals  which  command  our  admiration 
today,   which  work  is  not  excelled  by  modern   constructors  with 
advanced  knowledge  of  mathematics  and  mechanics. 

The  law  of  proportion,  as  applied  to  a  slab  or  beam  of  reinforced 
concrete,  may  be  stated  as  follows: 

4.  Variation   in  Strength   with   Thickness— -It  is  known  from 
elementary  principles  that  for  a  given  percentage  of  steel  and  a  given 
arrangement  of  reinforcement,  the  strength  of  a  slab  or  beam  in- 
creases directly  as  the  square  of  the  depth,  and  for  small  differences 
in  the  percentage  of  steel,  as  the  product  of  the  steel  area  times  the 
depth,  providing,  of  course,  the  steel  is  not  increased  to  such  an 
extent  that  the  steel  element  is  stronger  than  the  concrete  element. 

In  the  combination  of  concrete  and  steel,  it  should  be  observed 


54  VARIATION    OF    STRENGTH    WITH    SPAN 

that  as  between  the  two  elements,  the  steel  is  more  homogeneous, 
more  reliable  and  dependable  from  the  standpoint  of  uniformity 
of  strength.  Hence,  the  combination  should  be  made  in  such 
manner  that  should  failure  occur  it  would  occur  in  the  steel  and  not 
in  the  concrete,  and  when  this  principle  of  design  is  followed  out 
the  reliability  and  safety  of  the  structure  depends  on  the  steel  ele- 
ment, and  hence  no  greater  factor  or  margin  of  safety  is  needed  than 
in  structural  steel  work.  In  fact  there  would  be  less  uncertainty  in 
this  case  if  this  principle  were  carried  out  than  in  structural  steel 
work,  for  the  reason  that  in  structural  work  the  members  are  cut 
by  rivet  holes  and  there  is  less  dependence  to  be  placed  upon  the 
large  steel  shapes  so  wrorked  and  cut  than  in  the  case  of  rods  of 
uniform  section  and  not  so  nicked  or  cut. 

5.  Variation  in  Strength  with  Span — The  strength  decreases 
inversely  as  the  span  for  the  same  total  load  W,  and  the  same 
moment  of  resistance  of  steel  and  concrete.  If  the  strength  is  to  be 
compared  on  the  basis  of  a  unit  load  per  foot  of  span  length  then 
for  the  same  unit  load  the  strength  decreases  inversely  as  the  square 
of  the  span. 

These  fundamental  principles  of  proportion  enabled  the  earlier 
constructors  Coignet  and  Hennebique,  to  build  successfully  before 
the  development  of  the  theory  involving  the  relation  of  the  elastic 
properties  of  the  two  materials,  concrete  and  steel,  and  enabled 
Turner  to  successfully  build  many  great  structures  on  the  Mushroom 
flat  slab  system,  prior  to  the  development  of  a  rational  theory  based 
upon  the  elastic  properties  of  the  materials.  It  enabled  him,  not 
only  to  guarantee  the  strength,  but  also  to  guarantee  the  deflection 
of  his  structures  under  load. 

The  law  of  proportion  as  to  deflection  may  be  stated  as  follows: 

Within  practical  limits,  including  proper  percentages  of  steel, 
the  deflectiod  for  a  given  load  W  increases  as  the  cube  of  the  span  and 
decreases  inversely  as  the  product  of  the  steel  area  times  the  square 
of  the  depth  from  the  center  of  the  steel  to  the  top  of  the  slab  or  beam. 

These  proportionate  relations  are  sufficient  to  enable  the  practical 
constructor,  having  exact  knowledge  of  the  tested  strength  and 
deportment  of  a  reinforced  beam  or  slab  of  a  given  design,  to  design 
a  similar  beam  or  slab  for  a  larger  or  smaller  load  or  span  with 
certainty  as  to  the  result  which  can  be  obtained  with  the  same 
grade  of  workmanship  and  the  same  kind  of  concrete. 

The  method  of  proportion  applied  to  full  sized  structures  has 
this  advantage  over  all  other  methods.  It  takes  for  its  foundation 


TREATMENT  BY  PROPORTION  55 

or  starting  point,  the  tested  strength  of  a  member  approaching  in 
size  that  which  it  is  proposed  to  construct  and  a  comparison  is  made 
involving  a  narrower  range  for  the  application  of  the  principle  of 
proportionality  than  is  possible  where  the  theorist  undertakes  to 
develop  from  the  elastic  properties  of  a  minute  sample  or  unit  cube 
of  the  materials  employed  the  properties  of  a  full  sized  structure 
made  of  these  materials.  On  the  other  hand  the  method  of  propor- 
tion has  the  distinct  disadvantage  of  limitation  in  its  scope.  It 
cannot  be  applied  to  any  form  of  structure  which  differs  except 
within  narrow  limits  from  the  proportions  of  the  specimen  with 
which  it  is  compared,  and  hence  this  method  is  defective  as  compared 
with  a  general  solution  which  enables  broad  conclusions  to  be  drawn 
as  to  the  generic  type  under  consideration  rather  than  limited  con- 
clusions specific  to  one  form  only  of  the  genus. 

The  method  of  proportion,  based  as  it  is  on  elementary  relations, 
may  be  used  very  advantageously  to  verify  the  accuracy  of  more 
complex  and  scientific  methods  of  analysis.  The  relations  above 
outlined  follow  at  once  from  the  fundamental  principles  governing 
the  strength  and  flexure  of  beams  and  were  developed  in  substantially 
the  following  manner  by  Turner  in  his  treatise  on  Concrete  Steel 
Construction,  published  in  1909. 

6.  Theoretical  Treatment — A  slab  or  beam  supported  at  intervals 
either  on  points  or  on  walls,  if  loaded,  deflects  or  bends,  and  if  the 
load  is  excessive  the  concrete  cracks  first  from  the  lower  or  tension 
side  upward  in  a  plane  normal  to  lines  joining  the  supports.  Since 
the  reinforcing  metal  acts  by  tension  along  its  length,  it  is  evident  in 
general  that  so  far  as  the  steel  is  concerned,  whether  the  reinforce- 
ment is  in  single  layers  or  in  multiple  layers,  the  action  must  be 
similar  in  character  to  the  flanges  of  a  beam,  and  hence  the  strength 
of  the  beam  or  slab,  regardless  of  the  distribution  of  the  stress  in 
the  concrete,  depends  on  the  tensile  stress  in  the  steel.  The  mathe- 
matical expression  for  deflection  and  bending  would  be  of  identically 
similar  form  to  those  for  beams. 

W  =  the  total   load   on  the   beam   or   slab,   taken  for    convenience 

in  thousand  pound  units. 
L  =  the  span  in  feet. 
d  =  the  distance  from  the  top  of  the  concrete  to  the  center  of  steel 

in  inches. 

.4s  =  area  of  one  reinforcing  rod  in  square  inches. 
/s  =  the  intensity  of  actual  stress  in  the  steel  in  thousand  pound  units. 
2=  the  usual  sign  of  summation. 


56  FORMULAS    FOR    STRENGTH    AND    DEFLECTION 

M1  =  moment  of  resistance  for  stress  in  the  steel. 

A  =  deflection  at  the  center  of  beam  or  slab  for  any  load. 

(B)  =  a  coefficient  which  may  be  variable  or  constant  in  value,  to  be 

so  determined  experimentally  that  (B)  WL  shall  equal  MI. 
(D)  =  a  coefficient  similarly  obtained  for  deflection  formula. 

Then  by  the  laws  of  beams,  we  have  the  following  equations: 

Ml    =  (B)  W  L  =1   .85rf/3  2.4s (i) 

W  T3 

A     -tf»^ (2) 

SAscT 

In  formula  (1)  it  will  be  noted  that  .Sod  is  assumed  as  a  close 
approximation  to  the  effective  lever  arm  of  the  steel  jd,  or  the  dis- 
tance from  the  centroid  of  tension  to  the  centroid  of  compression 
in  the  beam  or  slab.  This,  of  course,  varies  slightly  with  different 
percentages  of  steel,  but  for  practical  purposes  it  may  be  assumed 
that  this  value  does  not  involve  material  error,  and  is  on  the  safe 
side. 

The  coefficient  (B)  for  the  simple  beam  is  £.  For  the  continuous 
beam  it  is  customary  to  take  this  as  1/12  at  the  support  and  1/16  at 
mid  span,  while  for  the  slab  supported  at  four  sides,  (B)  is  taken  as 
the  reciprocal  of  30,  and  where  the  reinforcing  metal  is  more  closely 
spaced  at  the  center  third,  the  reciprocal  of  36,  while  (B)  for  a  mush- 
room slab  is  taken  as  the  reciprocal  of  50.  It  is  assumed  in  these 
formulas  that  fs  =  13,  which  is  expressed  in  thousand  pound  units. 

The  deflection  A  of  slabs  will  follow  the  same  laws  as  the  deflec- 
tion of  beams  so  far  as  factors  depending  upon  W,  L,  and  I  are 
concerned,  and  will  consequently  be  equal  to  some  multiple  of 
WL3/EL  But  /  varies  as  2Asd2.  Hence  A  varies  as  WL3  /  2A8d2, 
which  is  expressed  above  in  equation  (2)  in  which  the  constant 
multiplier  (D)  is  to  be  determined  experimentally,  but  could  sup- 
posedly be  derived  analytically  in  case  a  sufficiently  general  theory 
should  be  developed  to  express  correctly  the  manner  in  which  it 
depends  upon  the  known  arrangement  and  properties  of  the  materials 
composing  the  slab. 

For  simple  beams,  (D)  is  taken  as  the  reciprocal  of  850  and  for 
continuous  beams  cast  integrally  with  a  heavy  slab,  as  the  reciprocal 
of  5,000. 

(D)  is  taken  as  the  reciprocal  of  10,000  in  the  slab  supported  on 
four  sides,  and  for  the  Mushroom  system  as  the  reciprocal  of  7,000. 


DERIVATION    OF    FORMULAS 


57 


The  application  of  the  principle  of  proportion  in  the  determina- 
tion of  the  working  stresses  is  based  on  the  assumption  (which  is 
on  the  side  of  safety)  that  steel  stress  under  working  load  is  pro- 
portional to  the  steel  stress  at  the  elastic  limit  of  the  steel  under  a 
load  which  would  produce  this  stress.  Now  the  elastic  limit  of 


Fig.  l-r>.      Mushroom  Column  Reinforcement  in  Curtiss  Building. 
Panels  approximately  16  ft. 


Fig.  16.     Slab  Reinforcement,  Mushroom  System,  in  a  wall 
bearing  building;  panels  about  16  ft. 

medium    open    hearth    steel    has  a  practically  fixed  value  which 
varies  little  and  which  can  readily  be  determined  by  test. 

Having  constructed  a  panel  or  beam  reinforced  with  this  known 
grade  of  metal,  by  loading  until  the  first  yielding  of  the  steel  occurs, 


58  EMPIRICAL    COEFFICIENTS 

we  then  know  the  steel  stress  under  the  applied  load,  and  may,  by 
proportion,  determine  closely  the  steel  tension  for  any  smaller 
working  load. 

The  elastic  limit  above  referred  to  is  the  limit  of  elasticity  of 
shape  as  determined  by  a  slowly  applied  load,  and  is  to  be  distin- 
guished from  rigid  proportionality  of  shape.  By  this  method  of 
investigation,  coefficients  (B)  may  be  determined  for  all  types  of 
construction.  Coefficients  (D)  are  figured  from  the  measured 
deflection  of  the  member  or  panel  tested.  This  method  of  deter- 
mining the  coefficients  used  is,  of  course,  limited  to  those  designs  in 
which  the  rational  method  is  followed  of  proportioning  the  structure 
so  that  the  steel  element  determines  the  safety  of  the  structure. 

No  formulas  for  strength,  based  on  the  elastic  properties  of  the 
materials  can  be  accepted  as  correct  unless  the  corrected  formula 
for  deflection  can  be  depended  upon  also.  In  other  words,  an 
elastic  theory  in  order  to  be  acceptable  must  demonstrate  its  accuracy 
by  agreement  of  the  entire  elastic  deportment  of  the  structure  to 
which  it  is  intended  to  be  applied  including  both  stresses  and  deflec- 
tions. The  determination  of  the  respective  coefficients  for  strength 
and  deflection  having  been  derived  independently  of  each  other, 
empirically  or  by  experiment,  their  general  agreement  can  be  es- 
tablished by  a  concordance  of  the  deflections  observed  with  those 
computed  in  structures  which  have  been  designed  for  strength, 
using  the  same  coefficients  (B),  for  both. 

We  have  noted  the  method  of  determining  the  coefficients  (B) 
and  (D)  for  a  specific  type  of  construction.  These  coefficients  can 
be  used  for  any  size  or  thickness  of  panel  or  percentage  of  reinforce- 
ment if  they  remain  constant  in  value  and  are  not  variables.  Their 
values  may  be  rendered  conscant  by  fixing  the  arrangement  of 
reinforcement  in  strict  proportion  to  the  sample  tested.  We  will 
illustrate  this  proposition  by  its  application  to  the  Mushroom  type 
of  reinforcement  shown  in  Figs.  15  and  16.  In  this  construction 
the  values  of  the  coefficients  are  made  constant  by  fixing  the  diameter 
of  the  Mushroom  head  and  width  of  belt  as  identically  or  approxi- 
mately the  same  fraction  of  the  span  of  the  test  panel  from  which 
the  coefficient  was  determined.  In  other  words,  if  the  diameter  of 
the  head  and  corresponding  width  of  belt  be  kept  within  the  limits 
of  7/16  to  1/2  the  distance  between  column  centers  in  the  case  of 
square  panels,  or  7/32  to  1/4  the  sum  of  the  long  and  short  spacing  in 
a  rectangular  panel,  then  the  coefficients  remain  practically  constant. 
Otherwise,  they  become  extremely  variable,  increasing  in  value 


RANGE    OF    APPLICATION 


59 


several  hundred  percent  as  the  width  of  belt  is  reduced  to  50  percent 
of  the  above  proportions.  From  this  statement  it  becomes  evident 
that  coefficients  of  this  character  must  be  applied  rigidly  to  similarly 


Fig.  17.     Interior  Stock  House  Hamm  Brewing  Company  Building,  showing  50  ton 
tanks  being  placed  in  position  covering  full  area  of  floor. 

proportioned  types  of  reinforcement  until  the  law  of  their  variations 
is  accurately  determined. 

L  in  the  formulas  for  bending  and  deflection  is  taken  as  the  longer 
direct  distance  between  column  centers,  the  shorter  direct  distance 


60  EXAMPLE    OF    COMPUTATION 

appearing  in  the  case  of  a  rectangular  panel  in  the  determination 
of  the  diameter  of  the  head  and  in  the  determination  of  the  load  W 
from  the  unit  load  per  square  foot. 

In  the  treatment  of  the  steel  area  it  may  be  noted  that  the  line 
of  weakest  section  at  failure  as  determined  experimentally,  cuts 
across  the  four  way  belts  practically  at  right  angles.  Hence  the 
resisting  moment  is  taken  at  this,  the  weakest  section,  as  the  elastic 
limit  of  the  steel  is  approached,  although  the  maximum  stress  in 
the  steel  does  not  occur  actually  at  this  section  on  the  diagonal  belts 
under  lesser  load. 

7.  Example  —  An  illustration  of  the  application  of  these  formulas 
is  given  herewith  for  the  Mushroom  system  in  the  floor  of  the  Hamm 
Brewing  Company  Stock  house,  illustrated  in  Fig.  17. 

Take  the  case  of  a  panel  of  the  Hamm  Brewing  Company's 
building,  shown  in  Fig.  17,  panel  22'10"  by  26/0'/  loaded  with 
four  tanks  W  in  diameter,  15'  high  full  of  water.  The  load  is 
equivalent  to  200  tons  of  uniformly  distributed  load.  The  floor  slab 
is  14"  thick  at  the  outer  edge  and  pitches  upwards  3"  to  the  center 
and  is  reinforced  with  twenty-five  f"  rounds  each  way.  Taking 
an  equivalent  depth  of  15J"  as  the  distance  from  center  of  steel  to 
top,  we  have  the  following  equation  : 


JL  x  .    144 

7000       (4x  25  x  .3)(15.25)2 

Another  panel  in  the  same  building,  20/10'/  by  20'8".  Same 
loading,  thickness  and  reinforcement. 

_j_x       (400)  (20.83)3          =  afulll/16,, 
7000       (4x25x  .3)(15.25)2 

These  figured  deflections  proved  exactly  equal  to  the  measured 
deflections  as  nearly  as  the  engineer  of  the  brewery  could  determine 
by  marking  the  same  with  a  knife  edge. 

We  will  take  another  case.  Test  of  the  State  Factory  Building 
at  Stillwater,  Minn.,  Mr.  C.  H.  Johnston,  Architect,  shown  in  Fig. 
18.  Size  of  panel  19'9"  by  20'8".  Thickness  of  slab  8".  Re- 
inforcement seventeen  f  "  rounds  each  way.  Test  load,  450  Ibs.  per 
square  foot  over  the  full  area. 

_     (180X20.  66V 

(7000)(4x  17x.ll)(7.25)2 
the  reported  deflection. 


RECTANGULAR    SLABS  61 

The  Hoffman  Building,  Milwaukee.  Test  load  142  tons.  Panel 
17'0"  by  16'8".  Reinforcement  seventeen  f"  rounds  each  way. 
Slab  8J"  (7|"  rough  and  1"  finish). 

(284)  (17)3 

(7000)  (4  x  17x".ll)(7.8)2 
the  measured  deflection. 


=  .437  =  7/16" 


Another  example:  Test  of  the  John  Deere  Plow  Companj^'s 
building  in  Omaha.  Fig.  18.  Panel  18'9"  square.  Reinforcement 
sixteen  |"  rounds  diagonally  and  fourteen  I"  rounds  directly  from 


Fig.  18.     Test  of  John  Deere  Plow  Company  Building,  Omaha,  Neb. 
550  pounds  per  square  foot. 


column  to  column.  7"  slab  in  rough,  with  strip  fill  added  later  about 
2J"  thick  and  I"  finish  floor  of  maple.  This  we  find  an  equivalent  to 
a  slab  of  about  8f"  concrete  as  far  as  deflection  is  concerned,  the  strip 
being  a  1:3^:4  mixture. 

(160)(18.75)3 


A  = 


(7000)  (6.6)  (8)' 


=  .356"  or  f",  the  measured 


deflection. 

8.  Slab  Supported  on  Girders — The  treatment  of  a  rectangular 
slab  supported  on  four  sides  and  cast  integrally  with  the  support- 
ing beams  by  the  method  of  proportion  will  also  be  illustrated. 


62 


FORMULAS    AND    COMPUTATION 


The  common  arrangement  of  such  slabs  is  shown  in  Figs.  19  and  20. 
For  square  slab  the  load,  of  course,  is  divided  equally  between  the  four 
supporting  beams  and  coefficient  (B)  is  taken  as  the  reciprocal  of  30 
where  the  rod  spacing  is  uniform,  and  as  the  reciprocal  of  36  where  the 
rods  are  spaced  twice  as  closely  for  the  middle  third  as  they  are 
for  the  outer  third.  The  coefficient  (D)  is  taken  as  the  reciprocal  of 
10,000  for  both  types.  In  the  case  of  a  rectangular  panel  on  the 
assumption  that  the  load  transferred  to  the  beams  is  in  proportion  to 
the  lengths  of  the  sides,  a  and  b,  a  mean  length  for  moment  and 
deflection  is  derived  by  taking  the  quotient  (a2-f&2)  /(a +6)  so 
that  the  formulas  become : 


IF 


4 


:-]. 


n 


n 


Fig.  19.  Fig.  20. 

Common  Type  Slabs,  supported  on  four  sides,  reinforced  two  ways. 


Ml  =  (B)  W(a2+b2)/(a+b)=  .85d/a  ?  AS/12 


9.     Computation  of  Deflection  Applied  to  Practical  Examples  — 

We  will  now  proceed  to  apply  these  formulas  to  two  cases  : 

First,  take  the  Minneapolis  Paper  Company's  building,  details 
shown  in  Fig.  21,  photograph  of  test  load  in  Fig.  22.  Slab  1"  in 
the  rough,  15'4"  by  21/6//  center  to  center  of  columns,  strip  fill  If" 
and  f"  finished  floor.  Now  taking  the  strip  fill  as  effective  for 
one  half  of  its  actual  thickness,  we  find  for  a  load  of  234,000  Ibs: 


m.S^lo.SV 
\21.5    +15.3  / 


.5    +15.3 

A  "  loTooo  x  234  x  775TTT)  x  (8?     '-  -30" 


COMPUTATION    OF    RECTANGULAR    SLAB 


63 


=  deflection  of  slab  at  center  as  measured  less  the  beam  deflection. 
Take  for  example  the  test  made  at  the  Smythe  block  at  Wichita, 
Kans.  Test  load  consisted  of  fifteen  tons  concentrated  at  the  center 
over  an  area  of  7  feet  square  equivalent  to  about  45,000  Ibs.  uniform 
load.  Size  of  slab  20'9"  by  24'9",  6J"  in  the  rough,  1}"  strip  fill. 
Reinforcement,  f "  rounds  5"  centers  for  the  central  third  of  the  panel 


ft! 


Fig.  21.     Details  of  Reinforcement  in  Panel  of  Minneapolis  Paper  Company  Building. 


each  way  and  8"  centers  outside  third  width  each  way.     Then— 

/2Q.752  x  24.75V 
1  \  45.5 / 

A  =  10,000 X(45)  9.x  7.25*  U"°r 

a  full  3/32"  the  deflection  measured. 

In  the  test  at  Wichita,  the  beam  deflection,  owing  to  the  small 
load  on  the  panel  was  practically  negligible,  and  does  not  need  to  be 
considered  in  arriving  at  the  true  slab  deflection. 

Test  of  the  Minneapolis  Knitting  Co.'s  Building: 

Slab  5J"  thick  with  If"  strip  fill,  panel  16'4"  x  15'8".  Re- 
inforcement f"  round  rods  4"  centers  each  way. 


64 


Fig.  22.     Photograph  of  Test  Load,  Minneapolis  Paper  Company  Building. 


SHORT    SPAN    SLABS    AND    ARCH    ACTION  05 


/16.33M-  15.66 

1  00  I 

•    (10.4)-x 


The  measured  deflection  agreed  identically  with  that  figured. 

This  formula  is  based  on  the  assumption  that  a  large  fraction  of 
the  internal  work  of  deformation  is  performed  by  lateral  action 
after  the  manner  of  a  uniform  continuous  plate  and  while  not  strictly 
accurate  it  is  in  much  closer  approximation  to  the  actual  condition  of 
stress  than  irrational  formulae  based  upon  a  distribution  of  stress 
and  load  in  proportion  to  the  fourth  power  of  the  respective  sides  as 
derived  in  the  irrational  treatment  on  the  basis  of  independent  beam 
strips  through  the  center  of  the  panel,  or  upon  an  inapplicable  ap- 
proximate solution  of  the  general  differential  equation  of  such  a  slab 
as  derived  by  Grashof  .  The  agreement  of  the  formula  for  deflection 
gives  closely  approximate  results  from  the  practical  standpoint  and 
its  mathematical  deviation  from  correct  values  will  be  discussed  later. 

The  method  of  design  by  proportion,  based  on  the  steel  stresses 
as  explained  in  the  preceding  pages,  presupposes  that  the  concrete 
element  resisting  compression  is  of  greater  strength  than  the  ten- 
sional  steel  element  as  it  should  be  in  conservative  design.  The 
limiting  steel  percentages  for  the  various  types  of  structure  will  be 
discussed  later  rather  than  under  this  present  heading. 

10.  Short  Span  Slabs  and  Arch  Action  that  may  be  Counted  upon 
in  Their  Use.  —  We  have  heretofore  discussed  at  some  length  long- 
span  slabs.  A  long  span  slab  will  be  defined  as  a  slab,  the  ratio  of 
whose  thickness  to  length  is  so  small  that  the  possibility  of  its  acting 
effectively  as  an  arch  is  eliminated. 

It  is  to  long  span  slabs  where  the  arch  action  is  negligible  that 
formulas  for  bending  apply  with  a  high  degree  of  precision.  When, 
however,  we  come  to  test  a  short  span  slab  which  is  part  of  a  con- 
tinuous floor  there  may  be  quite  a  large  amount  of  arching  in  the  slab 
by  which  the  load  is  carried  to  the  support  without  causing  tension 
in  the  steel  to  the  extent  that  it  would  do  provided  there  was  no  rigid 
skew-back  to  sustain  the  thrust.  Where  the  span  of  the  slab  does 
not  exceed  ten  times  its  thickness  it  is  permissible  and  good  practice 
to  figure  the  bending  moment  on  the  slab  at  WL/16,  and  thismoment 
is  to  be  increased  to  WL  /10  where  the  thickness  is  one-sixteenth  of 
the  span,  with  intermediate  values  for  intermediate  ratios  of  thick- 
ness to  span.  Where  the  span  is  more  than  sixteen  thicknesses  of  slab 
it  is  to  be  figured  as  heretofore  provided. 


66  FINISH    COAT    AND    STRIP    FILL 

11.  Value  of  Finish  Coat,  Strip  Fill  and  Wood  Floor  from  the 
Standpoint  of  Deflection — Many  engineers  have  an  idea  that  a  finish 
coat  or  strip  fill  or  the  like  cannot  act  in  connection  with  the  slab  to 
good  purpose  for  the  reason  that  the  bond  between  the  old  concrete 
of  the  slab  and  that  which  is  added  at  a  later  date  will  not  be  equal 
to  the  strength  of  the  original  concrete. 

While  this  is  true  to  some  extent,  nevertheless,  where  the  rough 
concrete  is  washed  off  and  scrubbed  with  a  steel  brush  and  then 
given  a  coat  of  neat  cement  grout  immediately  before  adding  the 
finish  coat  or  laying  the  strips  and  strip  filling,  the  concrete  is  nearly 
as  efficient  as  though  it  were  all  cast  at  the  same  time,  provided  that 
the  top  coat  or  strip  fill  is  given  a  reasonable  time  to  set  up  hard 
before  the  test  load  is  applied. 

A  If"  strip  fill  with  strips  16"  on  centers  and  a  £"  wood  floor 
generally  deflects  the  same  as  a  J"  or  I"  finish  coat  of  concrete. 
The  strip  fill  generally,  however,  if  the  strips  used  are  If'  will  some- 
what exceed  this  normal  thickness  since  it  is  impracticable  to  leave  the 
top  surface  of  the  rough  slab  perfectly  level,  and  we  count  as  a  rule 
that  the  actual  thickness  of  the  nominal  1  j"  strip  fill  will  not  be  less 
than  2i"  in  the  center  of  the  slab  tho  it  may  be  a  little  less  around 
the  columns  if  the  columns  are  poured  in  accordance  with  our  standard 
practice. 

If  we  assume  that  the  bond  between  the  finish  coat  and  the  old 
concrete  is  an  even  30  percent  of  the  strength  of  the  original  concrete 
we  would  still  have  a  very  large  factor  of  safety  in  view  of  the  great 
area  of  the  slab  to  take  care  of  the  horizontal  shear  between  the  two 
layers.  This  is  a  fact  which  is  generally  disregarded  by  those  who 
are  dealing  with  reinforced  concrete.  If  a  slab  the  depth  of  which 
has  been  increased  by  perhaps  20  percent  by  strip  fill  and  the 
finish  be  figured  on  the  basis  of  the  actual  thickness  of  the  rough  slab 
only,  a  surprisingly  high  degree  of  strength  will  apparently  be  de- 
veloped on  this  basis  by  test,  but  a  more  conservative  computation 
taking  into  consideration  the  actual  value  as  about  one  half  of  this 
added  thickness  will  estimate  the  construction  at  its  true  worth. 

For  strip  fill,  where  strength  becomes  an  object,  identically  the 
same  grade  of  concrete  should  be  used  as  in  the  slab,  instead  of 
the  weak,  indifferent  mud  filling  of  cinders  or  natural  cement  or  brown 
lime  which  is  sometimes  employed.  Further,  by  using  Portland 
cement  in  the  strip  fill,  the  contractor  will  find  that  this  filling  hard- 
ens up  and  dries  out  much  more  promptly  than  any  mixture  of 


STRIP    FILL 


natural  cement,  brown  lime,  or  Portland  cement  and  lime,  thus 
permitting  the  finished  floor  to  be  laid  at  an  earlier  date  without 
danger  of  having  the  hard  wood  swell,  buckle  and  rise  up  from  the 
cleats  to  which  it  is  nailed  by  reason  of  the  moisture  absorbed  from 
the  uncured  filling. 

In  a  case  where  lime  was  used  with  an  idea  of  economy  in  a 
building  completed  in  the  fall  which  the  owners  were  in  a  hurry  to 
occupy,  this  filling  dried  very  slowly,  seeming  to  have  an  affinity  for 
moisture,  and  when  the  finished  floor  was  laid  it  swelled  so  as  to 
buckle  up  in  places  eighteen  and  twenty  inches  high,  due  to  the  swel- 
ling of  the  boards  longitudinally,  while  laterally  the  swelling  of  this 
kiln  dried  maple  was  over  fifteen  inches  in  a  width  of  fifty  feet.  Six 
widths  had  to  be  taken  out  of  the  floor,  and  the  floor  taken  up  and 
entirely  relaid.  The  saving  in  first  cost  of  fill  thus  proved  very 
expensive  in  the  end. 


68 


CHAPTER  III 
BEAMS 

1.     Elastic  Properties  of  Materials,  Concrete  and  Steel.    In  the 

design  of  a  composite  structure,  such  as  a  reinforced  concrete  beam 
or  member  by  an  elastic  theory,  it  is  necessary  to  know  the  relative 
stresses  under  like  deformations.  These  will  depend  upon  the  ratio 
of  the  moduli  of  elasticity  of  the  two  respective  materials. 

For  safe  design  we  need  to  know  the  range  or  limits  between  which 
the  ratio  assumed  holds  true.  For  the  steel,  Young's  modulus  is 
#s  =  3xl07.  The  elastic  limit  of  medium  steel  may  be  taken  as 
35,000  pounds  per  square  inch  and  for  hard  steel  50,000  pounds  per 
square  inch.  (See  Standard  Specifications.) 

The  resistance  which  concrete  offers  to  crushing  is  variable  as  we 
have  pointed  out  in  our  analysis  of  the  strength  of  concrete,  and 
depends  upon  the  proportions  of  the  mixture,  the  character  of  the 
sand,  gravel  and  stone,  as  well  as  the  conditions  of  hardening  and  age 
of  the  concrete.  The  form  and  size  of  test  specimens  also  influences 
the  apparent  strength.  The  age  of  the  specimens  has  a  marked 
effect  upon  the  strength  as  well  as  upon  the  modulus  of  elasticity. 

As  to  increase  of  strength  with  age,  Morsch  quotes  tests  in  con- 
nection with  a  bridge  erected  over  the  Danube  at  Munderkingen, 
with  one  part  cement,  two  and  one  half  parts  sand  and  five  parts 
pebbles.  Test  cubes  developed  the  following  strength : 

After  28  days,    average  strength  compression  3613  Ibs.  per  sq.  in. 

"       5  months,     "  "  "  4722     "     "     "     « 

"       2  yrs.  8  mo."        compressive  strength  7396     "     "     "     " 

"       9  years,  compressive  strength 8107  "     "     " 

These  values  are  materially  higher  than  the  average  value  of 
broken  stone  concrete,  which  may  be  accounted  for  by  the  excellent 
quality  of  sand  and  the  hardness  and  grade  of  pebbles  used. 

Since  the  strength  of  concrete  increases  with  time,  it  is  permissible 


STRENGTH  AND  ELASTIC  PROPERTIES  OF  CONCRETE  69 

to  use  higher  working  stresses  when  making  an  addition  to  an  old 
building  constructed  of  good  concrete. 

2.  Tensile  Strength.    Results  of  tensile  tests  are  more  variable 
than  those  of  compression.     In  most  cases,  tensile  tests  are  made  on 
mortar  specimens;  that  is  those  composed  of  cement  and  sand  only. 
Few  tests  have  been  made  on  ordinary  concrete  specimens  with 
coarse  aggregate.     The  latter  exhibit  less  tensile  resistance  than  the 
specimens  of  mortar. 

In  general  it  may  be  stated  that  the  tensile  strength  of  concrete 
may  be  taken  as  having  a  value  between  one  tenth  and  one  twelfth 
of  the  ultimate  compressive  strength. 

3.  Elasticity  of  Concrete.    It  is  impossible  to  assign  a  definite 
value  for  the  modulus  of  elasticity  of  concrete  since  all  of  the  factors 
entering  into  the  breaking  strength  influence  its  elastic  behavior 
and  make  it  difficult  to  compare  the  results  obtained  by  different 
observers. 

Concrete,  unlike  steel,  has  no  definite  elastic  limit,  the  stress 
strain  curve  of  a  block  when  first  tested  in  compression,  being  a 
curved  line  from  the  beginning,  due  in  part  to  shrinkage  stresses 
induced  in  the  process  of  hardening.  Considering  only  the  stress 
strain  curve  obtained  the  first  time  it  is  loaded  we  might  say  that  the 
modulus  is  not  practically  a  constant  quantity  like  that  for  steel  but 
has  only  an  instantaneous  value  which  varies  for  any  given  specimen 
with  the  load. 

Concrete  further  differs  from  steel  in  taking  permanent  sets 
under  very  light  loads,  and  if  these  permanent  sets  are  not  deducted 
from  the  total  deformation  under  gradually  increasing  load  the 
result  does  not  represent  the  true  elastic  deformation.  This  was 
pointed  out  by  Professor  Bach  of  Stuttgart  in  1895.  He  found  for  a 
given  maximum  loading  of  less  than  half  the  ultimate  strength  that 
repeated  loading  eliminates  the  permanent  set  and  gives  a  fairly 
constant  modulus  for  subsequent  loadings  not  exceeding  this  max- 
imum. 

As  illustrating  the  variation  of  the  modulus  of  elasticity  with  the 
age  of  the  specimen,  the  results  noted  in  the  following  table  according 
to  Morsch,  are  of  interest: 


70 


STRESS    STRAIN    CURVE    OF    CONCRETE 


TESTS  OF  OLD  CONCRETE  FOR  MODULUS  OF 
ELASTICITY 


Unit  Stress 


Three  Months 
Old 


Two  Years 
Old 


Remarks 


1US./  ill 

I   "   Ec 

Ec 

1223  .  1 

3655000 

1048.2 

3741000 

871.9 

2973000 

3812000 

.1   697.0 

3072000 

3869000 

Average  of 

1   523.4 

3158000 

3954000 

three  tests 

|   435.2 

3229000 

3983000 

o   348.5 

3342000 

4025000 

260.3 

3428000 

4068000 

173.5 

3613000 

4125000 

3769000 

4330000 

0 

22.8 

3271000 

4836000 

44.1 

2944000 

4495000 

65.4 

2845000 

4423000 

One 

g    88.2 

2759000 

4409000 

Single 

'§   109.5 

2489000 

4381000 

test 

H   130  .  8 

4310000 

each 

153.6 

4310000 

174.9 



4281000 

196.3 



4239000 

The  accompanying  diagram  Fig.  23,  gives  a  fair  idea  of  the  stress 
strain  curve  plotted  from  test  results  at  one  and  5  months.  Fig.  24 
shows  the  stress  strain  curve  arrived  at  by  repeated  loading. 

When  the  load  is  applied  gradually,  the  shortening  of  the  specimen 
which  is  at  first  small,  increases  more  and  more  rapidly  as  the  load 
increases  as  shown  in  the  concave  curve  Fig.  24,  plotted  with  the 
applied  loads  as  ordinates  and  the  deformation  as  abscissas.  As  the 
loading  is  gradually  removed  the  curve  YO'  takes  a  convex  form  and 
shows  a  permanent  set  00'  on  a  horizontal  axis.  On  again  applying 


ELASTIC    DEPORTMENT    AT    DIFFERENT    PERIODS 


71 


the  same  load,  the  new  stress  strain  curve  starting  from  the  new  origin 
0'  is  still  of  concave  form  looked  at  from  the  right  j  ust  as  the  original 
curve  OY  was,  but  to  a  lesser  degree  and  for  the  same  load  as  at  Y  the 
point  Y'  shows  a  smaller  relative  set  than  the  set  00' .  On  unloading 


\ 

xl 


£337 


174 


/4-3Z 


//fa 


OOOZ 


oovt 


w* 


.00*4 


Fig.  23. 


Stress  Strain  Curves  in  Compression  from  1:2:4    Cylinders,  thirty  days  and 
one  hundred  fifty  days  old,  respectively. 


again  the  origin  is  moved  slightly  to  0"  '.  With  several  successive 
applications  and  removals  of  the  same  load,  the  origin  is  continually 
removed  slightly  to  the  right,  but  the  movement  becomes  less 
and  less  until  there  is  no  additional  permanent  set.  The  permanent 


72 


CONCRETE    UNDER    REPEATED    LOADING 


set  of  concrete  appears  then  to  be  in  a  great  measure  reached  under 
the  first  loading  and  for  subsequent  applications  of  the  same  load 
the  concrete  acts  more  and  more  nearly  as  a  perfectly  elastic  material. 


Deformsf/oo 

Fig.  24.     Stress  Strain  Curves  in  Compression  of  Concrete  under  Repeated  Loading. 

4.  Concrete  Beams.  Where  comparatively  large  sections  of 
metal  are  used  for  the  purposes  of  directly  resisting  tensile  stress 
due  to  bending  the  ratio  of  the  modulus  of  elasticity  of  the  steel  to 
that  of  concrete  is  for  practical  purposes  generally  taken  as  one  to 
fifteen,  and  the  tensile  strength  of  the  concrete  is  entirely  neglected 
This  is  for  the  usual  1:2:4  concrete.  For  a  rich  mix  such  as  1:1J:3 
this  ratio  is  sometimes  taken  as  one  to  ten  or  twelve. 

As  pointed  out  in  the  historical  sketch,  engineering  opinion  has 
crystalized  in  the  adoption  of  the  linear  law  of  distribution  of  stress 
for  purposes  of  computation  of  beams  and  slabs  and  the  assumptions 
involved  in  this  modification  may  be  stated  as  follows : 

(a)  Adhesion  between  the  concrete  and  steel  shall  be  sufficient 
to  make  the  two  materials  act  together. 

(b)  The  steel  is  to  take  all  direct  tensile  stress. 

(c)  The  stress  strain  curve  of  the  concrete  in  compression  is  a 
straight  line  for  the  range  of  working  stress. 

(d)  A  plane  cross-section  of  an  unloaded  beam  will  still  be  plane 
after  bending. 

(e)  The  material  in  the  beam  will  obey  Hooke's  law  in  that 
stress  is  proportional  to  strain. 

From  the  above  it  follows  that  unit  deformations  of  the  fibers  at 


MODULUS    OF    ELASTICITY    OF    CONCRETE    IN    BEAMS  73 

any  section  are  proportional  to  their  linear  distance  from  the  neutral 
surface,  and  the  unit  stress  in  the  fiber  at  any  section  of  the  beam  is 
proportional  to  the  distance  of  the  fiber  from  the  neutral  surface. 
The  linear  law  above  stated  is  the  basis  of  all  practical  formulas  of 
flexure  except  some  which  have  been  developed  for  reinforced  con- 
crete beams  applicable  to  the  conditions  as  failure  is  approached 
rather  than  the  condition  for  safe  loads  or  for  safe  working  stress. 

The  Joint  Committee  (American  Society  Civil  Engineers,  etc.) 
on  Concrete  and  Reinforced  Concrete,  recommended  that  calcula- 
tions be  made  with  reference  to  working  stresses  and  safe  loads  rather 
than  with  reference  to  ultimate  strength  and  ultimate  loads,— an 
endorsement  of  customary  practice  of  experienced  builders  in  this 
respect.  It  also  endorsed  current  practice  with  regard  to  the  modulus 
of  elasticity  and  to  commonly  accepted  formulas  for  beams  which 
are  reproduced  herewith  as  follows: 

5.  Modulus  of  Elasticity.     ''The  value  of  the  modulus  of  elas- 
ticity of  concrete  has  a  wide  range,  depending  on  the  materials  used, 
the  age,  the  range  of  stresses  between  which  it  is  considered,  as  well 
as  other  conditions,     It  is  recommended  that  in  computations  for 
the  position  of  the  neutral  axis  and  for  the  resisting  moment  of  beams 
and  for  the  compression  of  concrete  in  columns  it  be  assumed  as : 

(a)  One-fifteenth  of  that  of  steel,  when  the  strength  of  the  concrete 
is  taken  as  2200  Ibs.  per  sq.  in.  or  less. 

(b)  One-twelfth  of  that  of  steel,  when  the  strength  of  the  concrete 
is  taken  as  greater  than  2200  Ib.  per  sq.  in.  or  less  than  2900  Ib. 
per  sq.  in.,  and 

(c)  One-tenth  of  that  of  steel,  when  the  strength  of  the  concrete  is 
taken  as  greater  than  2900  Ib.  per  sq.  in. 

Altho  not  rigorously  accurate,  these  assumptions  will  give  safe 
results.  For  the  deflection  of  beams,  which  are  free  to  move  longi- 
tudinally at  the  supports,  in  using  formulas  for  deflection  which  do 
not  take  into  account  the  tensile  strength  developed  in  the  con- 
crete, a  modulus  one-eighth  of  that  of  steel  is  recommended." 

6.  Formulas  for  Reinforced   Concrete  Construction  as  recom- 
mended by  the  Joint  Committee. 

(a)     Standard  Notations 
1.     Rectangular  Beams. 

The  following  notation  is  recommended : 
fa  =  Tensile  unit  stress  in  steel, 
/c  =  Compressive  unit  stress  in  concrete, 
Es  =  Modulus  of  elasticity  of  steel, 
Ec  =  Modulus  of  elasticity  of  concrete, 


74  STANDARD    NOTATION 

M  =  Moment  of  resistance,  or  bending  moment  in  general, 

Ms  for  steel,  Mc  for  concrete, 
A  =  Steel  area, 
b  =  Breadth  of  beam, 
d  =  Depth  of  beam  to  center  of  steel, 
k  =  Ratio  of  depth  of  neutral  axis  to  effective  depth  d, 
z  =  Depth  of  resultant  compression  below  top, 
j  =  Ratio  of  lever  arm  of  resisting  couple  to  depth  d, 
jd  =  d — z  =  Arm  of  resisting  couple, 
p  =  A  /bd  Steel  ratio  (not  percentage). 

2.  T-Beams. 

6  =  Width  of  flange, 
b'  =  Width  of  stem, 
t  =  Thickness  of  flange. 

3.  Beams  Reinforced  for  Compression. 

A'  =  Area  of  compressive  steel, 
pf  =  Steel  ratio  for  compressive  steel, 
fsf  =  Compressive  unit  stress  in  steel, 
C  =  Total  compressive  stress  in  concrete, 
C'  =  Total  compressive  stress  in  steel, 
d'  =  Depth  to  center  of  compressive  steel, 
z  =  Depth  to  resultant  of  C  and  C". 

4.  Shear  and  Bond. 

V  =  Total  shear, 
y  =  Shearing  unit  stress, 
u  =  Bond  stress  per  unit  area  of  bar, 
o  =  Circumference  or  perimeter  of  bar, 
2o  =  Sum  of  the  perimeters  of  all  bars. 

5.  Columns. 

A  =  Total  net  area, 
As  =  Area  of  longitudinal  steel, 
Ac  =  Area  of  concrete, 

P  =  Total  safe  load. 

(b)     Formulas 

1.     Rectangular  Beams. 

Position  of  neutral  axis, 

k=  V  2  pn  +  (pn)2—pn     (1) 


FORMULAS    FOR    BEAMS 


75 


Arm  of  resisting  couple, 


j=l—  k  .......................................  (2) 

(For  /a=  15,000  to  16,000,  and  /c  =  600  to  650,  j  may  be  taken  at  {.) 


K  — -tc 


M 


•2     ' 


...(3) 

..(4) 


Steel  ratio,  for  balanced  reinforcement, 
1  1 


/c 

2.     T-Beams. 

Case   I.     When  the  neutral   axis  lies  in  the  flange:     Use  the 
formulas  for  rectangular  beams. 

Case  II.     When  the  neutral  axis  lies  in  the  stem. 

The  following  formulas  neglect  the  compression  in  the  stem: 


K"b'->i 
Position  of  neutral  axis, 


kd  = 


2nA 


+  bt2 

4-  2bt 


-.(6) 


76  FORMULAS    FOR    BEAMS 

Position  of  resultant  compression, 

_3  kd— 2t    t       (7) 

~  2kd—  t'3 

Arm  of  resisting  couple, 

jd  =  d—z (8) 

Fiber  stresses, 

M 

f*  =  Ajd  -'(9) 

Mkd  f      k 

/c  =  -7 ST     -£r    k     (10) 

btlkd-itjjd 

(For  approximate  results,  the  formulas  for  rectangular  beams 
may  be  used.) 

The  following  formulas  take  into  account  the  compression  in  the 
stem;  they  are  recommended  where  the  flange  is  small  compared  with 
the  stem: 

Position  of  neutral  axis, 


-,     i          \/    —    n/\jt'j.±.     |     \  <_/         L/     /  (/  |    /(/^.L     |     \  iy         iy     /  L  \  Ti'./i  "j"  10     0    )t 

6'  \  &'          /  b' 

(ID 

Position  of  resultant  compression, 


t(2kd—t)b+(kd—t)2bf 


(12) 

kd—t)b  +  (kd—t)2b' 

Arm  of  resisting  couple, 


jd  =  d~z  ........................................  (13) 

Fiber  stresses, 


c~   [(2  kd—  t)  bt+(kd—t)2  bf]  jd 

3.     Beams  Reinforced  for  Compression. 
Position  of  neutral  axis, 

k=  V  2  n~(7;+7/  d'/d)"  +  n2(p+p'f—  n  (p+pf)  ............  (16) 


SHEAR    AND    BOND 


77 


Position  of  resultant  compression, 

_  ^  k3d+2  p'nd'(k—d'/d)  (17) 

/c2+2  p'n  (k—d'/d) 

Arm  of  resisting  couple, 

jd  =  d—z (18) 


Fiber  stresses, 

J  C  ~mr 


1— A; 


(19) 


. . (20) 


fs  =  nfc(k-d'/d)/k..  ..(21) 

4.     Shear,  Bond  and  Web  Reinforcement. 

In  the  following  formula,  2o  refers  only  to  the  bars  constitut- 
ing the  tension  reinforcement  at  the  section  in  question,  and  jd  is 
the  lever  arm  of  the  resisting  couple  at  the  section. 


For  rectangular  beams, 
V 


V 


.  .  (22) 


.  .  (23) 


(For  approximate  results,  j  may  be  taken  at  |.) 
The  stress  in  web  reinforcement  may.be  estimated  by  the  following- 
formulas  : 


78  DETERMINING    MOMENT    IN    BEAMS 

Vertical  web  reinforcement, 


Web  reinforcement  inclined  at  45°  (not  bent-up  bars), 

Vs 

P  =  0.7vd  .......................................  (25) 

in  which  P  =  stress  in  single  reinforcing  member,  V  =  amount  of  total 
shear  assumed  as  carried  by  the  reinforcement,  and  s  =  horizontal 
spacing  of  the  reinforcing  members. 

The  same  formulas  apply  to  beams  reinforced  for  compression  as 
regards  shear  and  bond  stress  for  tensile  steel. 

For  T-beams, 

,  =  JL  ....................................  ......  (26) 

b'jd 


(For  approximate  results,  j  may  be  taken  at  |.) 

5.     Columns. 

Total  safe  load, 

P  =  fc  (Ac+nAs)=fcA  (\+(n—l)p)  .................  (28) 

Unit  stresses, 


A  (l  +  (n—  l)p)  ...............................  (29) 


7.  Determining  Moment.  In  case  the  steel  element  in  a  rein- 
forced concrete  beam  is  weaker  than  the  concrete,  the  determining 
resistance  is  that  of  the  steel,  and 


(a) 

If,  on  the  other  hand,  the  beam  be  over  reinforced,  the  determining 
moment  is  that  of  the  concrete,  and 


(b) 

For  approximate  calculations  it  will  be  sufficiently  correct  to  assume 
.7  =  0.85  and  A;  =  0.40,  these  being  fair  average  values  for  steel  per- 
centages from  0.75  to  1.25.  Equations  (a)  and  (b)  then  become 

Ms  =  0.85  Adf3 
and  M  =  0.1 


POSITION    OF    NEUTRAL    AXIS    AND    EFFECTIVE    DEPTH 


79 


r 

/a 

\ 

\ 

s\ 

K-  Curves 

\ 

Y 

\ 

.26 

Z& 
JO 

T? 

Y 

\N 

ss 

\ 

s  \ 

s  \ 

. 

\ 

X 

\ 

^ 

Y 

x 

<^>x 

X 

x 

^ 

^ 

^^ 

v 

X 

^ 

x^ 

^\ 

c^. 

.38 

40  :\ 

44 

\ 

\^ 

" 

^-^ 

X^ 

"^ 

s^ 

^ 

-^ 

\ 

Q*  r 

•"\. 

• 

^^^ 

^ 

\^ 

^ 

- 

-\ 

~-» 

-—  -^ 

^_ 

"^-~- 

^. 

-^_ 

' 

"^-^ 

^ 

5/9- 

"-^ 

=- 

i_4 

<^en\ 

&    0,9    f. 

r/ 

Of 

'    /. 

?  /, 

J   /. 

0. 

5'/.a 

/   a, 

s  a 

6   a 

7    0, 

4     / 

5_/ 

(     /.7     /i 

l^A 

Fig.  25.     j  and  A;  curves. 


80 


STEEL    RATIO    FOR    GIVEN    VALUES    OF  /c    AND  /s 


More  exact  values  of  j  and  k  for  various  values  of  p  and  n  may  be 
obtained  from  the  accompanying  j  and  k  curves  Fig.  25,  which  have 
been  computed  and  plotted  from  formulas  (1)  and  (2)  of  Section  6. 
Also  values  of  fa  and  /c  plotted  in  accordance  with  equations  (a) 
and  (b)  above,  are  shown  in  Fig.  26a  and  26b. 


/7O 


Percevtege 


Fig.  26a. 


The  theory  of  the  flexure  of  reinforced  concrete  beams,  assumes 
that  for  all  practical  purposes  they  may  be  assumed  to  obey,  with 
sufficient  accuracy,  the  so-called  straight  line  theory  of  stresses  and 
strains,  a  theory  under  which  according  to  Hooke's  law  the  stress  is 
proportional  to  the  strain  and  varies  directly  as  the  distance  from 
the  neutral  axis.  But  the  assumption  is  more  or  less  inaccurate 
because  the  stress  strain  curve  of  the  concrete  in  the  compression 
zone  is  not  a  straight  line  since  the  rate  of  deformation  is  greater 
where  the  stress  is  larger.  This  is  equivalent  to  saying  that  the 
modulus  of  elasticity  Ec  of  concrete  in  compression  becomes  smaller 
the  larger  the  unit  stress  /c  becomes  and  vice  versa.  This  variation 


DEEP  AND  SHALLOW  BEAMS  COMPARED 


SI 


of  the  modulus  Ec  as  well  as  any  small  initial  permanent  deformation 
of  the  concrete  will  cause  some  deviation  of  reinforced  concrete  beams 
from  perfectly  elastic  flexure.  But  this  deviation  will  be  less  ap- 
preciable the  deeper  the  beam,  because  any  slight  increase  of  deforma- 
tion in  the  upper  fiber  will  have  less  effect  upon  the  sharpness  of 


O.5% 


f.5% 


of    &  e/n  fore  erven  f 

Fig.  2Gb.     Diagram  of  Steel  Percentages  for  Different  values  of  M    :    bd* 

bending  in  a  deep  beam,  and  consequently  have  less  effect  in  causing 
cracks  or  checks  in  the  concrete  on  the  under  side  of  it  than  in  a 
shallow  beam.  In  other  words,  the  deviations  of  concrete  from 
perfect  elasticity  have  less  effect  the  deeper  the  beam,  and  their 
effect  in  increasing  sharpness  of  curvature  and  cracking  of  thoconcrete 
will  be  more  pronounced  the  more  shallow  the  beam. 

How  these  deviations  affect  the  position  of  the  neutral  axis  and 
the  sharpness  of  the  bending  may  be  made  evident  from  Fig.  27 
which  is  intended  as  a  representation  on  a  large  sale  of  the  deforma- 
tions, etc.,  occurring  in  a  unit  length  of  two  different  beams  of  the 


82 


BEAMS    WITH    EQUAL    STEEL    STRESS 


same  depth,  the  two  beams  being  superimposed  on  each  other  in 
the  Fig.  27  to  assist  in  the  comparison. 

Let  the  two  beams  have  different  percentages  of  reinforcement, 
but  be  so  loaded  that  the  unit  steel  stress  fs  is  the  same  in  both.  A 
larger  load  will  be  necessary  to  produce  the  same  unit  steel  stress 
in  the  beam  with  the  larger  percentage  of  steel,  but  the  actual  unit 
deformation  of  the  steel  will  be  the  same  in  both  beams,  viz : 
A0  =  e=fs/Es. 

In  beam  No.  1,  with  the  lighter  load  and  smaller  value  of  the  steel 
ratio,  assume  that /c  the  compressive  stress  in  the  extreme  fiber  is  so 
moderate  that  the  concrete  in  compression  may  for  practical  purposes 


Q'&BJ 

'    /    I  /'V 


Fig.  27. 

be  regarded  as  obeying  Hooke's  law.  Then  the  line  AB  by  its  horizon- 
tal distance  from  00'  at  different  levels  represents  the  relative  unit 
deformations  at  those  levels.  But  by  Hooke's  law  these  distances 
would  also  represent  unit  stresses  when  measured  in  a  suitable  scale, 
because  stress  is  deformation  multiplied  by  modulus  of  elasticity 
which  latter  is  taken  to  be  constant  in  this  case.  In  beam  No.  1 
with  light  load  and  small  steel  ratio  pi  the  total  tension  in  the  steel 
and  the  total  compression  in  the  concrete  will  each  be 


The  corresponding  neutral  axis  at  Ci  and  point  of  application  of 
compressive  stress  TI  will  be  nearer  the  top  of  the  beam  the  smaller 
B1  is. 

In  beam  No.  2  with  heavier  load  and  larger  steel  ratio,  p2,  while 
AO  the  elongation  is  unchanged  the  line  of  deformations  will  assume 
some  new  position  AB2,  and  if  the  compressions  near  the  top  surface 
are  large  enough  in  this  case  to  make  the  modulus  of  elasticity  less 
than  it  is  at  points  nearer  the  neutral  axis,  the  horizontal  distances 


"See  Merriman's  Mechanics  of  Materials  p.  273. 


VARIATION    OF    Ec   WITH    DEPTH    AND    STEEL    RATIO  83 

of  this  line  of  deformations  AB2  will  no  longer  also  correctly  repre- 
sent the  stresses  to  scale.  Those  near  the  top  of  the  beam  will  be 
smaller  when  plotted  to  the  same  scale  as  in  No.  1.  But  that  would 
reduce  the  area  T2  between  it  and  00'  if  the  neutral  axis  atC2  remains 
fixed.  In  fact,  however,  the  total  steel  tension 

T2  =  A2fs  =  A2  e  #9  =  area  0'  C2B2 

is  a  fixed  quantity  and  the  neutral  axis  must  be  moved  to  some 
lower  position  C2  in  order  that  the  total  compression  represented 
by  the  area 

T2  =  0'C2B2  =  0'C2B2  =  T2f 

may  remain  constant.  It  is  evident  then  that  there  is  first  a  lower- 
ing of  the  position  of  the  neutral  axis  from  Cl  to  C2  by  reason  of  the 
increase  of  pi  to  p2  and  next  a  lowering  of  it  from  C2  to  C2  by  reason 
of  the  decrease  of  the  modulus  of  elasticity  Ec  under  large  values  of 
the  unit  stress  /c.  This  explains  more  fully  why  high  values  of  /(. 
should  be  avoided  in  shallow  beams. 

On  the  other  hand,  deviations  of  concrete  from  perfect  elasticity 
are  less,  the  less  the  actual  compressive  unit  stress  /c  acting  upon  the 
concrete.  In  other  words,  concrete  under  the  smaller  stresses 
behaves  more  nearly  like  perfectly  elastic  material.  An  effective 
method,  therefore,  of  reducing  the  sharpness  of  bending  and  conse- 
quent exaggerated  tendency  of  shallow  beams  to  check  and  crack  is 
to  make  the  compressive  stress  /c  in  the  concrete  small  compared 
with  fs  the  given  stress  in  the  steel.  This  is  equivalent  to  making 
the  stress  ratio  fa  //c  larger  for  shallow  beams  than  for  deep  beams, 
as  is  the  practice  among  experienced  builders. 

There  is  still  another  way  of  stating  the  consideration  which 
leads  to  the  adoption  of  small  values  of  /„  for  shallow  beams.  It  is 
desirable  to  limit  deflections  under  working  loads  to  a  figure  not 
much  in  excess  of  L  /1000.  With  fa  given,  the  adoption  of  large 
values  of /s  //c  will  make  the  compressions  in  the  concrete  so  moderate 
as  to  prevent  excessive  deflections  even  tho  there  should  be  some 
small  initial  deflection  due  to  non-elastic  compression  of  the  con- 
crete. Values  of  /s  //„  from  16  to  35,  for  values  of  n=  12  and  n  =  15 
which  have  been  computed  from  formula  (5)  section  6,  have  been 
plotted  in  the  accompanying  Fig.  28. 

Since  the  steel  element  in  the  combination  is  more  dependable 
than  the  concrete  from  the  standpoint  of  uniformity  of  strength,  the 
safety  of  the  structure  is  made  by  experienced  builders  to  depend  on 
the  steel.  In  order  to  effect  this  the  working  strength  of  the  concrete 
should  be  taken  at  a  smaller  fraction  of  its  ultimate  strength  than 


84 


STEEL    RATIO    FOR    GIVEN    VALUES    OF  fs/fc 


the  working  strength  of  the  steel  is  of  its  ultimate  strength.  For  a 
1 :2 :4  mix  650  pounds  per  square  inch  is  a  safe  working  stress  to  resist 
compression  in  concrete  arising  from  bending.  Both  tension  and 
compression  are  developed  in  concrete  by  flexure  and  by  bond  shear. 
The  resistance,  however  which  it  offers  to  tensile  stress  is  small 
compared  with  that  which  it  offers  to  compression.  Forty  pounds 
per  square  inch  is  a  safe  value  of  the  working  tensile  resistance. 


jp 

I 


Jt> 

\ 

\ 

\ 

\ 

33 

\    \ 

\ 

\ 

\ 

\ 

\ 

\ 

£9 

\     *• 

s 

23 

\ 

\ 

\ 

.-^ 

\ 

V3 

\ 

5 

\ 

\ 

\ 

\ 

\ 

x^ 

V 

\ 

in 

x 

X 

x^ 

/a 

\ 

:O 

x^ 

/7 

r^e 

^"^^ 

^^ 

/z 

^ 

s-s^ 

\^ 

if 

1    •>, 

•%^ 

t^. 

O.3%    O.4       GS        O.6        O.7        O.0        dff        f.O        /./         X>2         /.3        f.4        f.S 

Percenf&ge    of  /3e/nforcemenf 

Fig  28.     Limiting  per  cent  of  Steel  for  Different  values  of  /s//>'. 


Now  if  a  beam  is  to  depend  for  its  stability  upon  the  stress  in 
the  steel,  that  stress  must  not  exceed  a  certain  assigned  value  depend- 
ent upon  the  quality  of  the  steel,  or  what  amounts  to  the  same 
thing  a  given  quality  of  steel  must  not  suffer  a  stress  or  a  corres- 
ponding elongation  at  any  point  in  excess  of  an  assigned  value.  In 
order  to  compare  beams  of  different  depths  and  the  same  assumed 
maximum  stress  /s  or  elongation  e  of  steel,  draw  two  plane  sections 
of  the  beam  under  consideration  at  right  angles  to  the  neutral  axis 
and  at  a  distance  of  one  unit  apart.  Before  bending  occurs,  the  two 
sections  are  parallel  to  each  other,  but  after  bending  they  make  the 


DEEP    AND    SHALLOW    BEAMS    WITH    EQUAL    STEEL    ELONGATIONS  Sf> 

same  elementary  angle  A  0  with  each  other.  Draw  thru  one  extremity 
of  the  unit  of  length  at  the  neutral  axis  (which  has  been  unchanged 
in  length  by  bending)  a  plane  parallel  to  the  section  at  the  other 
extremity  of  the  unit  length.  It  consequently  makes  an  angle 
A  0  with  the  original  plane  section  at  this  extremity.  Any  horizontal 
shearing  deformation  may  be  disregarded  in  this  comparison  because 
it  will  affect  both  sections  to  practically  the  same  amount.  The 
unit  elongation  of  the  steel  due  to  the  bending  between  these  two 
unit  sections  will  be  e  =  (l-k)dA  0. 

This  investigation  is  made  upon  the  assumption  that  e  has  a 
value  which  is  constant  and  the  same  for  different  beams,  but  with 
the  proviso  that  under  this  steel  elongation  neither  the  shearing- 
distortion  nor  the  compression  of  the  concrete  anywhere  shall  exceed 
permissible  limits,  questions  which  will  have  to  be  separately  in- 
vestigated since  they  depend  on  the  steel  stresses  in  too  complex  a 
manner  to  be  readily  introduced  into  consideration  at  the  same  time 
with  the  effect  of  the  constancy  of  the  steel  stresses. 

Now  other  things  being  equal  A  0  decreases  as  p,  the  percentage 
of  the  reinforcement,  increases;- i.  e.  A0  =  c/p  where  c  is  an  ex- 
perimental constant  whose  value  is  dependent  upon  the  grade  of 
concrete,  etc.  Substitute  this  value  of  A  0  in  the  previous  expres- 
sion, then 

e/c=(l-fc)rf/p=  (1~*)L 
pN 

in  case  d  be  assumed  to  be  some  known  fraction  1  /N  of  the  span  L, 
i.  e.  or  d  =  L  /N. 

It  thus  appears  that  the  last  member  of  this  equation  will  be  found 
to  be  an  experimental  constant  for  reinforced  beams  of  the  same 
span  and  grade  of  concrete;  and  in  case  the  numerical  value  of  this 
constant  be  determined  for  any  given  beam  not  liable  to  excessive 
deformation  at  the  center,  it  will  have  the  same  value  for  a  beam  of 
different  depth,  span,  and  percentage  of  steel,  provided,  as  before 
stated  that  sufficient  resistance  to  compression  and  diagonal  tension 
be  supplied. 

For  example,  assume  /8=  13,000  and  /c  =  650,  then  /8//c  =  20. 
Referring  to  the  curve  for  /s  //c  Fig.  28  the  corresponding  steel 
ratio  is  p  =  0.0105,  and  taking  the  corresponding  value  of  A- 
from  the  k  curve,  it  appears  that  (1 — k)=  0.573.  It  is  known  by 
experience  that  a  beam  whose  depth  is  1/12  of  the  length  should 


STEEL    RATIO    FOR    GIVEN    RELATIVE    DEPTHS 


have   this   amount   of  reinforcement,   or   N=12   when  p  =  0.0105. 

(l-k}L        0.573  L 
Hence 


.   „  T 
— = 4 . 5o  L 


pN  0.0105x12 

is  the  constant  for  such  beams. 

The  same  curve  shows  that  for/s//c  =  26,  p  =  0.007  and  (1-fc) 
=  0.635;  hence,  using  these  and  the  constant  4.55,  we  find  J¥  =  20. 


at     a?     as     ae     /.o 
Perceqtege    of 


Again,  for/B//c  =  32,  p   =0.005  and  (1-A:)  =0.68  and  .¥  =  30. 

The  large  values  of  /s  //c  which  have  been  assumed  above  for  the 
shallow  beams  N  =  2Q  and  N  =  30,  reduce  the  working  stress /c  and 
the  steel  ratio  p  below  the  values  for  .¥=  12  in  accordance  with  good 
practice.  The  accompanying  diagram  Fig.  29  gives  values  of 
L  /d  =  N  for  usual  values  of  p  computed  from  the  equation  N  = 
(l-/c)/4.55  p  andn=15. 


SHEAR    FOR    GIVEN    STEEL    RATIO  87 

Now  the  total  tension  T  in  the  steel  at  mid  span  of  a  simple  beam  is 
T  =  WL  /Sjd.     But  T  =  bdpf»  in  which  fs  is  the  unit  steel  stress  at 

WL 

mid    span,  hence  /s  =  -    — =-. 
8  jbd'p 

But  the  identical  elongations  e  of  both  the  concrete  and  the  steel 
at  mid  span  may  be  written 

e  =/8  /Es  =/c  /Ec,     or  /8  //c  =  n 


and  /c  =          2 

8  j&crpn 

in  which  fc'  is  the  apparent  direct  tensile  stress  in  the  extreme  fiber  of 
the  concrete  at  mid  span  as  shown  by  its  elongation  e  while  in  con- 
tact with  the  reinforcement.  The  so  called  apparent  stress  /</  may  or 
may  not  correspond  to  an  actual  stress  of  some  considerable  amount. 
It  is  used  here  simply  as  another  way  of  expressing  the  actual  elonga- 
tion e.  The  experiments  of  Considere*  show  that  concrete  when 
well  reinforced  may  remain  intact  under  elongations  not  only  far  in 
excess  of  any  possible  for  concrete  without  reinforcement  but  in  fact 
remain  intact  under  elongations  several  times  as  great.  The  reason- 
ing here  employed  is  however  entirely  independent  of  any  question 
of  actual  checking  or  not,  for  /c'  =  eEc  is  simply  a  convenient  unit  of 
comparison  computed  as  the  product  of  elongation  and  modulus. 

Next  obtain  the  shearing  stresses  and  the  diagonal  tension  in 
the  concrete  at  the  extremity  of  a  simple  beam.  The  total  horizontal 
shear  between  a  unit  of  length  of  the  reinforcement  and  the  concrete 
is  such  that  a  segment  of  the  beam  lying  between  two  vertical  planes 
which  are  one  unit  apart  is  held  in  equilibrium  by  the  total  vertical 
shear  \W  acting  with  the  arm  unity  and  the  total  horizontal  shear  S 
acting  with  the  arm  jd. 

Hence  %W  =  S  jd,  or  S  =  \W  /jd. 

This  makes  the  unit  horizontal  shear  on  any  horizontal  plane 
below  the  neutral  axis 


provided  the  total  shear  in  a  unit  length  be  regarded  as  uniformly 
distributed  thruout  the  breadth  6  of  the  beam.  This  is  equal  to 
the  unit  diagonal  tension  at  the  end  of  the  beam  which  is  produced 
by  the  shear  alone. 

Hence  s  =  4  pn  fc  /N 

Take  /c  =  1000  and  p  =  lo  then  s  =  60,000  p  /N 

*Experimental  Reasearches  on  Reinforced  Concrete,  McGraw  Pub.  2nd  Ed.,  p.  224 


88  SHEAR    AND    RESULTANT    DIAGONAL    TENSION 

By  using  corresponding  values  of  p  and  .¥  given  previously  we  find 

p  N  s 


.0085 

16 

32 

.009 

15 

36 

.0095 

14 

40.7 

.01 

13 

46 

Plotting  the  values  of  s  corresponding  to  the  assumed  values  of 
p  it  appears  that  s  will  reach  a  safe  limiting  value  of  40  Ibs.  per  sq. 
inch  when  p  =  .  0094  nearly  and  when  the  span  is  somewhat  more 
than  fourteen  times  the  thickness,  or  L  /#  =  JV=  14  .25,  as  may  also 
be  seen  from  the  diagram  Fig.  29.  Beams  more  shallow  than  this  will 
have  smaller  values  of  s,  but  deeper  beams  where  Ar<14.25  will 
require  reinforcing  to  resist  diagonal  tension  at  the  ends,  when  there 
is  a  working  stress  of  16,000  Ibs.  per  sq.  inch  on  the  steel  at  mid  span. 

Reinforcement  for  the  purpose  of  increasing  the  resistance  to 
diagonal  tension  consists  of  diagonal  or  vertical  rods  toward  the  ends 
of  the  beam.  The  reinforcement  may  be  introduced  in  such  amount 
as  to  make  unit  steel  stresses  greater  or  less  at  mid  span  than  at  the 
ends.  The  total  resultant  diagonal  tension  at  any  point  of  the  beam 
per  unit  of  length  of  the  reinforcement  is  compounded  of  the  total 
direct  stress  in  the  steel  and  the  total  shearing  stress  per  unit  of 
length  of  the  steel,  and  is 


in  which  the  letters  T  and  S  designate  the  total  direct  steel  tension 
due  to  bending  and  the  total  shear  ait  the  point  considered  respect- 
ively and  do  not  signify  as  previously  the  total  tension  at  mid  span 
and  the  shear  at  the  end.  The  inclination  i  of  this  resultant  tension 
R  to  the  horizon  is  found  from  the  expression 

cot  2t  =  J  T  /S 

The  value  of  R  at  any  point  of  the  span  may  be  readily  constructed 
graphically  as  shown  in  the  accompanying  diagram  Fig.  30,  in  which 
the  ordinates  of  the  parabola  called  the  T7  curve  represent  the  total  steel 
tension  at  any  point  of  the  span  due  to  the  load  band  and  the  or- 
dinates of  the  straight  line  called  the  S  curve  represent  the  total  band 
shear  per  unit  of  length  of  the  steel  at  any  point.  Then  at  any 
point  P  the  resultant  R  =  PP'  is  constructed  and  laid  off  vertically 
in  two  segments  PP"  =  \  T,  and  the  hypothenense 


*  See  Merriman's  Mechanics  of  Materials,  Page  273. 


DIAGONAL    TENSION 


89 


The  ordinates  of  the  locus  of  P'  give  the  total  diagonal  tension  R. 

The  total  vertical  force  in  the  beam  per  unit  of  length  of  span 
which  must  be  resisted  by  vertical  reinforcement  or  tension  in  the 
concrete  or  both  is  S  =  V  /jd  as  given  by  equation  (24)  Section  6. 
This  is  shown  by  the  S  curve.  At  a  safe  value  of  forty  pounds  per 
square  inch  of  vertical  tension  in  the  concrete,  the  safe  vertical 
resistance  of  the  concrete  per  unit  of  length  of  span  is  406  pounds, 
Draw  a  horizontal  line  on  the  diagram  thru  some  point  Q  at  this 
height  above  P.  Then  vertical  or  diagonal  reinforcement  is  necessary 
at  all  points  of  the  span  where  the  S  curve  lies  above  QQ  and  the 


Fig.  30. 


total  amount  of  the  tension  to  be  resisted  by  the  vertical  reinforcement 
at  any  point  per  foot  of  span  is  represented  by  the  vertical  distance 
of  the  S  curve  above  QQ  at  that  point. 

It  is  evident  that  the  point  at  which  a  beam  will  first  fail  by 
diagonal  tension  depends  upon  S — 406  as  compared  with  the  amount 
and  distribution  of  the  reinforcement,  and  lies  at  the  point  where 
the  maxium  unit  stress  occurs.  This  may  occur  at  any  point, 
depending  upon  the  amount  of  the  vertical  steel.  In  beams  with 
the  reinforcing  rods  turned  upward  at  the  ends  of  the  beam  and 
securely  anchored  there  the  point  is  usually  removed  to  some  distance 
from  the  ends. 

The  treatment  just  given  of  the  vertical  stress  in  th&  concrete 
assumes  that  the  verticals  consist  of  stirrups  or  the  like  at  some 
considerable  distances  apart  horizontally,  say  10"  or  more. 


90  WEB    REINFORCEMENT 

The  case,  however,  is  different  if  the  required  vertical  steel 
consists  of  rods  or  wires  so  near  together  as  to  prevent  checking  or 
cracking  of  the  concrete  until  the  vertical  steel  has  a  working  stress 
of  13,000  to  16,000  pounds  per  square  inch.  In  that  case,  diagonal 
tension  failure  need  not  occur  before  the  concrete  has  an  elongation 
e  corresponding  to  an  apparent  tensile  stress  of  at  least  600  pounds 
per  square  inch,  instead  of  40  pounds;  but  in  that  case  the  vertical 
steel  should  be  designed  to  resist  the  entire  vertical  tensions. 

If  in  addition  to  distributing  the  vertical  steel  along  the  span  so 
completely  as  to  fully  reinforce  the  concrete  vertically,  some  of  the 


Fig.  31.     View  Showing  Beam  Failure  by  Diagonal  Tension  near  the  End. 


longitudinal  steel  be  run  parallel  to  the  neutral  axis  so  as  to  fulh' 
reinforce  the  concrete  below  the  neutral  axis  longitudinally  as  well 
this  will  introduce  coaction  of  the  vertical  and  horizontal  steel  in 
such  a  way  as  to  materially  reduce  the  steel  stresses  in  the  web,  in 
the  same  manner  as  occurs  in  the  steel  stresses  in  slabs.  This  is 
the  explanation  of  the  striking  results  obtained  by  the  beam  designs 
of  Maciachini,  and  of  Cottangin  as  shown  in  Fig.  44. 

The  theoretical -deduct  ions  which  have  been  reached  in  the  pre- 
ceding pages  may  be  confirmed  by  reference  to  a  great  mass  of  test 
data.  It  will  be  sufficient,  however,  at  present  to  refer  to  certain 
of  the  tests  reported  in  Bulletin  No.  197,  of  the  University  of  Wis- 


See  Marsh,  Reinforced  Concrete,  2nd  Edition,  Page  77. 


WITHE Y  S    TEST    BEAMS 


91 


Dl 


1    1 


A 

7,  /  Corr  Sr/rrups 


•rnk!  i  !  r 


-/-/^N^P-KJ  i  i 

-1—  -1 — ^e 


riTHNvj  Mini 


1 1  rpu    i 
'HIJ  l>i< 


Mesh,nq    /  Mesh 


IIW/re—     Fl 


r  — t 


^IL 


COT  Bars' 


rz 


sn 

sMJJ 


-^0&~ 


IO-0 


Fig.  32.      Reinforcement  of  Test  Beam  and  Cracks 


92 


DIAGONAL    TENSION 


cousin,*  from  which  Figs.  31,  32  and  33  have  been  taken.  They 
show  the  details  of  the  yielding  and  failure  of  several  beams  with 
the  checking  of  the  concrete  as  well  as  the  amount  and  arrangement 
of  the  reinforcement.  The  beams  represented  are  all  T-beams 
supported  at  the  ends  with  practically  one  percent  of  reinforcement, 
and  N  =  L/d<9.  The  top  flange  of  the  beams  of  these  beams 
afforded  sufficient  resistance  to  make  any  moment  failure  occur  by 
yielding  of  the  steel  in  the  bottom  of  the  beam.  Moreover,  there 
was  sufficient  reinforcement  against  diagonal  tension  to  prevent 
failure  of  that  kind  in  the  beams  of  series  D,  E,  and  F,  but  not 
enough  in  Series  G.  Every  failure  by  yielding  of  the  steel  at  mid 
span  caused  an  amount  of  deflection  and  a  sharpness  of  bending 
that  crushed  the  concrete  in  the  flange.  The  first  tension  cracks 


-44 


Gl 

.3-4'- 


i i i   i    i 


i   i  i 


}  £  Cor r  Bars) 


\3\f\*  \' 


A         ^Qttrrup  trotf  A 

Fi6.  33.     View  Showing  Reinforcement  and  Cracks  Test  Beams,  Diagonal  Failure 


in  the  middle  third  began  to  be  visible  on  the  bottom  at  a  stress  of 
12000  to  15000  Ibs.  per  square  inch  in  the  steel.  In  Series  F,  the 
reinforcement  against  diagonal  tension  contained  no  verticals  such  as 
occurred  in  Series  D  and  E,  but  wire  mesh  was  used  instead  as  repre- 
sented. The  cracks  show  that  the  beams  of  series  F  while  actually 
failing  by  direct  tension  in  the  steel  were  nevertheless  appreciably 
nearer  failure  by  diagonal  tension  than  were  those  of  series  D  and  E. 
It  is  stated  the  mean  tensile  strength  of  6  inch  test  cylinders  of  con- 
crete cast  at  the  same  time  as  the  beams  was  187  Ibs.  per  square 


*Tests  on  Plain  and  Reinforced  Concrete,  Series  of  1907,  by  Morton  Owen 
Withey,  C.  E. 


FAILURE    BY    DIAGONAL    TENSION  93 

inch  and  that  the  beams  began  to  crack  by  diagonal  tension  when 
the  unit  vertical  tensile  stress  computed  by  the  formula 

S  =  ±W  /jdb 

reached  a  mean  value  of  179  Ibs.  per  square  inch  for  all  of  them. 
This  is  in  good  agreement  with  the  direct  tensile  strength  of  the  con- 
crete as  just  quoted,  since  it  differs  from  it  by  less  than  five  per 
cent.  But  were  the  steel  well  spaced  at  such  short  intervals  along 
the  span  as  to  fully  reinforce  the  web  vertically  as  was  done  by 
Maciachini  previously  referred  to,  and  were  web  steel  introduced 
all  the  way  from  the  ends  to  the  loads  where  the  shears  began,  we 
should  expect  to  find  the  concrete  take  much  larger  deformations 
and  apparent  stresses  than  this  without  cracking. 

It  will  be  noticed  that  these  beams  were  all  loaded  with  equal 
concentrated  loads  at  the  one  third  points  of  the  span.  Such  a 
loading  makes  the  moment  and  shear  curves  very  different  from  those 
given  in  Fig.  30.  The  moment  curve  will  be  horizontal  thruout 
the  middle  third.  It  also  makes  the  shear  curve  zero  in  the  middle 
third  and  horizontal  in  the  end  thirds.  These  beams  were  con- 
sequently especially  liable  to  diagonal  tension  failure  at  some  dis- 
tance from  the  ends  where  the  anchorage  of  horizontal  steel  at  the 
ends  exerts  no  effect. 

Such  a  failure  is  shown  in  Fig.  31  where  a  J  inch  stirrup  is  broken 
just  below  the  arrow,  and  as  a  consequence,  because  the  concrete 
was  unable  to  resist  the  vertical  stress  the  crack  then  extended  along 
the  horizontal  steel  at  the  bottom  and  along  the  flange  at  the  top. 

It  should  be  noted  that  the  age  of  these  beams  at  the  time  of 
testing  was  only  28  days,  at  which  time  the  resistance  of  the  concrete 
to  shear  and  diagonal  tension  was  probably  not  more  than  40  percent 
of  the  ultimate,  a  fact  that  would  be  likely  to  make  their  behavior 
when  fully  cured  materially  different  from  that  exhibited  at  the  time 
of  test,  so  far  as  shearing  and  diagonal  tension  are  concerned. 

8.  Discussion  of  the  Elastic  Properties  of  Beams  and  Assump= 
tions  involved  in  the  Preceding  Theory.  In  discussing  the  elastic 
properties  of  concrete  it  was  shown  that  the  modulus  of  elasticity  of 
the  concrete  is  not  constant  for  different  loads  and  further  that  the 
modulus  changes  with  the  age  of  the  concrete,  it  being  only  f  as  great 
at  the  age  of  thirty  to  forty  days  as  it  is  at  the  age  of  two  or  more 
years,  and  furthermore  that  while  this  modulus  is  usually  considered 
the  same  for  tension  and  compression,  this  is  open  to  some  question. 
Accordingly  a  reasonable  approximation  to  the  conditions  which 


94  DISCUSSION    OF    BEAM    THEORY 

occur  in  bending  when  the  building  is  first  ready  for  occupancy  is 
attempted  and  in  nearly  all  building  codes  the  ratio  of  the  modulus  of 
elasticity  of  steel  to  concrete  in  bending  is  assumed  at  1  to  15.  A 
difference  in  this  ratio  would  affect  the  position  of  the  neutral  surface 
to  some  extent  and  the  effective  lever  arm  to  a  still  smaller  extent; 
as  shown  by  comparison  of  the  j  and  k  curves  in  the  diagram  in 
Fig.  25.  The  assumption  of  the  value  of  n  as  15  for  bending  is 
accordingly  on  the  safe  side,  and  the  error  involved  is  not  great.  On 
the  other  hand,  this  divergence  of  practical  conditions  from  the  as- 
sumptions used  in  the  computation  do  not  justify  a  high  degree  of 
mathematical  precision  in  the  work  of  practical  design,  for  if  the 
computations  are  carried  to  a  degree  of  nicety  unwarranted  by  the 
accuracy  or  agreement  of  the  assumption  with  practical  conditions 
it  is  a  mere  expenditure  of  time  without  commensurate  results. 
Accordingly,  it  may  be  stated  that  the  approximate  formulas  for 
beams  and  slabs  are  sufficiently  accurate  for  practical  purposes. 

In  T-beams  where  the  beam  is  integral  with  the  slab,  it  is  cus- 
tomary to  assume  a  width  of  slab  not  exceeding  four  times  the  slab 
thickness  as  forming  a  part  of  the  compressive  flange  of  the  beams. 
This  assumption,  of  coarse,  is  conservative.  It  is  evident  that  the 
compression  in  the  outside  edge  of  that  portion  of  the  slab  which  is 
regarded  as  useful  section  is  less  than  portions  nearer  the  axis  of  the 
beam,  and  that  the  slab  beyond  this  imaginary  division  is  also 
restrained  in  compression,  the  condition  approaching  what  has  been 
designated  as  the  "twilight  zone"  between  exact  knowledge  and 
conjecture  as  to  the  actual  conditions.  Evidently  in  a  case  like  this, 
exact  computations  beyond  the  limits  of  accuracy  of  the  assumption 
is  a  waste  of  time  and  the  effective  depth  jd  of  a  T-beam  may  be  for 
practical  purposes  determined  at  once  by  the  assumption  of  (l-j)d 
=  |  t  without  material  error  when  d  lies  between  2.5  t  and  4  t. 

In  case  of  the  rectangular  beam  with  f  to  1J  percent  reinforce- 
ment, the  assumption  of  jd  =  .  85  d  is  sufficiently  accurate  for  practical 
purposes,  and  for  percentages  of  steel  less  than  f  percent  the  assump- 
tion of  jd  =  .  9d  is  sufficiently  accurate. 

>  >'  The  case  of  the  doubly  reinforced  beam  is  one  which  the  designer 
rarely  is  called  upon  to  make  use  of.  In  it  the  compression  steel 
should  preferably  be  about  2J"  to  3"  from  the  top  surface  of  the  beam. 
Unless  the  percentage  of  tensile  reinforcement  is  very  high,  say  3 
percent  or  more,  and  the  compressive  reinforcement  very  low,  say 
less  than  0.75  percent,  the  neutral  plane  is  nearer  the  compressive 
steel.  Assuming  d'  to  equal  d  /10  when  p'  is  2  percent,  it  is  nearer 


TENSILE    AND    COMPRESSIVE    REINFORCEMENT  95 

the  compressive  steel  for  all  values  of  p'  '.  Thus  it  follows,  since  the 
unit  stresses  in  the  compressive  and  tensile  reinforcements  are  as 
the  distances  of  these  reinforcements  from  the  neutral  plane,  that 
the  unit  stress  in  the  compressive  steel  is  for  these  percentages  less 
than  that  in  the  tensile  steel.  For  very  rough  approximate  compu- 
tations, taking  n  =  15  and  the  average  value  of  j=  .85,  k=  .45,  we 
have 

1.17M  =  M 

'   pbd2  (.19-10.  5V)  bd2 

The  above  formula  for  fa  is  a  fair  approximation.  The  formula 
for  fc  with  different  percentages  of  steel  is  by  no  means  a  close  ap- 
proximation. 

A  much  more  satisfactory  method  of  computation  is  as  follows  : 
From  equations  (20)  we  find 


(19a) 

To  determine  /c  we  cannot  assume  an  arbitrary  value  of  k  in  this 
equation  since  that  would  be  tantamont  to  assuming  that  the  amount 
of  compressive  steel  would  make  no  difference  in  /c,  hence  this  equa- 
tion cannot  be  used  as  an  approximate  method  of  determining  fc, 
but  it  may  be  employed  to  determine  /c  in  an  accurate  manner  by 
plotting  the  curve  from  the  values  of  n(l-k)/k  for  different  per- 
centages of  tensile  and  compressive  steel  from  which  we  may  derive 
fc  by  dividing  /s  by  the  value  taken  from  the  diagram. 

The  accompanying  Fig.  34  shows  the  curves  of  different  percent- 
ages of  tensile  steel  for  different  percentages  of  compressive  steel 
reinforcement  noted  at  the  bottom  of  this  Fig.  The  values  of  j  are 
given  at  the  left  of  the  diagram  and  the  values  of  /s  -:  -fc  for  the 
different  percentages  of  tensile  and  compressive  reinforcing  steel 
given  at  the  right. 

9.  Classification  of  Beams:  This  is  based  upon  the  manner 
in  which  the  beam  is  supported  : 

A  simple  beam  is  one  which  is  merely  supported  at  the  ends, 
and  its  mathematical  treatment  is  based  upon  the  consideration 
that  the  beam  is  free  at  the  end  to  turn  and  that  the  supports  offer 
no  resistance  to  rotation.  Such  a  beam,  does  not,  of  coarse,  exist  in 
practice,  but  all  beams  which  rest  upon  supports  and  are  not  rigidly 
restrained  or  have  only  a  small  degree  of  restraint  at  supports  are 
treated  from  the  practical  standpoint  as  simple  beams  and  figured  as 
such. 


96 


COMPRESSIVE    STEEL 


Concrete  beams  or  slabs  which  are  apparently  continuous  over 
supports  but  which  have  reinforcing  metal  at  the  bottom  thruout, 
offer  so  slight  resistance  to  negative  moment  at  supports  that  they 
are  treated  on  the  theory  of  predominant  action  as  simple  beams 
unless  their  depth  be  sufficient  and  the  longitudinal  restraint  offered 
by  the  construction  such  that  may  be  treated  somewhat  on  the  arch 
principle.  A  slab  supported  on  parallel  walls  and  reinforced  in  one 
direction  is  merely  a  wide  beam  and  with  reinforcement  in  the  bottom 
thruout  is  to  be  treated  merely  as  a  wide  simple  beam. 


o   a/  az  0.3  c&  as  0.6   0.7  as  a9  /.o   //   /*   /.3  /.+  /.s  /.6  /.  7  /.s 


Fig.  34. 


Where,  however,  a  beam  is  continuous  thruout,  and  rigidly  built 
into  and  integral  with  a  series  of  columns,  and  suitable  reinforce- 
ment is  provided  at  the  top  of  the  beam  over  the  support  and  extend- 
ing outward  to  the  line  of  inflection,  and  then  thruout  the  bottom 
of  the  beam,  we  have  a  true  continuous  beam  in  which  the  bending 
moments  over  the  support  follow  the  laws  of  continuous  beams 
except  as  they  are  modified  by  the  rigidity  of  their  integral  union  or 
connection  with  the  columns. 

The  effect  of  this  monolithic  connection  is  to  cause  the  deport- 
ment of  the  beam  to  approach  more  and  more  nearly,  for  all  spans, 
to  the  condition  of  a  continuous  beam  extending  through  an  indefinite 


CONTINUOUS    BEAMS 


97 


number  of  spans;  in  other  words,  to  cause  the  moment  over  the 
support  for  uniform  load  to  become  WL  /12  and  the  moment  at 
mid  span  WL  /24.  An  end  span  of  the  series,  however,  except  in 
heavy  warehouse  construction  will  not  receive  this  full  degree  of 
restraint.  In  a  heavy  warehouse  with  the  large  columns,  26  inches 
and  over  in  diameter,  this  degree  of  restraint  is  for  practical  purposes 
fully  secured,  but  with  smaller  columns  it  may  be  less,  and  its  amount 
is  to  be  determined  approximately  by  the  designer  from  comparison 
of  the  relative  rigidity  of  the  columns  and  beams,  so  that  in  the  case 
of  light  columns  the  end  spans  should  be  somewhat  more  heavily 
reinforced  for  moment  at  mid  span. 

The  case  of  an  unloaded  span  with  both  adjacent  spans  fully 
loaded  is  not  uncommon  in  a  warehouse  and  this,  too,  must  be  provided 
for. 


Span  18 'd"  t 


Fig.  35.     Continuous  Beam,   Turner  System. 


Where  the  construction  consists  cf  beams  in  but  one  direction 
with  the  slab  spanning  from  beam  to  beam  and  with  insufficient 
metal  parallel  to  the  beam  in  the  slab  to  fully  reinforce  the  beam  to 
resist  the  negative  moment  under  the  circumstances  stated,  which  may 
result  from  the  excess  of  live  load  stress  over  and  above  the  dead 
load  stress,  the  beam  should  be  treated,  in  determining  the  central 
moment,  as  continuous  for  dead  load  only  and  as  a  simple  beam  for 
the  live  load. 

The  preferable  arrangement,  however,  is  the  provision  of  beams 
in  both  directions  from  column  to  column  where  I: earn  and  slab 
constructions  are  used,  making  the  floor  a  true1  monolith  or  a  natural 
concrete  type.  For  this  type  of  construction  with  ordinary  spans, 
beam  reinforcement  consisting  of  say  five  rods  arranged  as  indicated 
in  Fig.  35  is  preferable,  in  which  the  beam  rods  consist  of  two  which 
extend  thruout  the  length  of  the  beam  at  the  bottom  and 
into  the  adjacent  span,  two  which  are  bent  up  from  the 
quarter  point  to  the  top  of  the  beam  and  extend  over  into 


98  CONTINUOUS    BEAMS 

the  adjacent  span,  and  one  which  while  extending  into 
adjacent  spans  slopes  gradually  from  the  top  of  the  support  to  the 
bottom  of  the  beam  near  the  center  of  the  span. 

An  arrangement  of  this  kind,  after  the  manner  of  the  Bollman 
truss,  furnishes  liberal  provision  for  shear  at  the  support,while  the  in- 
clined rods,  under  bending  strain  resist  shear  thru  their  inclination 
at  the  support.  It  is  only  necessary  to  figure  the  moment  over  the 
support  as  WL  /1 2  and  provide  therefor  by  the  cross  section  of  the 


Continuous  Beam,  Hennebique  System. 


six  rods  crossing  it  while  the  four  rods  in  the  bottom  of  the  beam 
and  the  five  rods  at  the  center  are  ample  for  all  possible  conditions 
of  loading.  The  lap  of  rods  at  the  support  and  beyond  the  support 
both  at  the  top  and  bottom  render  sudden  failure  or  collapse  practical- 


Fig.  37.     Cross  Section  of  Beam  and  Slab. 

ly  impossible  after  the  concrete  has  had  even  a  few  hours  under 
normal  temperature  conditions  in  which  to  harden. 

In  addition  to  enhanced  safety  there  is  very  material  economy 
in  such  an  arrangement,  since  compared  with  beams  of  constant 
section,  the  continuous  beam  is  more  than  five  times  as  stiff  and  one 
and  one  half  times  as  strong  as  the  simple  beam  having  the  same 
cross  section  of  metal  thruout  the  bottom  of  the  beam  as  the  con- 
tinuous beam  has  at  the  top  over  the  support. 

Fig.  36  shows  the  Hennebique  continuous  beam  which  has  an 
enviable  record  from  the  standpoint  of  safety  by  virtue  of  the  liberal 
lap  of  reinforcement  and  stirrup  verticals  employed. 


INDEPENDENT    BEAM    CONSTRUCTION  99 

The  fact  that  concrete  is  well  adapted  to  be  placed  in  a  mono- 
lithic mass  renders  continuous  construction  the  natural  type  to  use 
for  the  reason  that  it  combines  the  highest  degree  of  safety  with  the 
maxium  of  stiffness  and  economy. 

Even  tho  settlement  of  the  supports  should  occur  and  the  con- 
crete should  check  by  reason  thereof,  the  well  designed  continuous 
floor  does  not  become  dangerous  and  unsafe.  So  long  as  the  concrete 
is  hard  and  rigid,  the  checked  segments  can  take  compression,  and 
the  steel  while  the  bond  is  intact  can  furnish  the  full  resistance  to 
tension  which  was  originally  figured  upon  disregarding  the  direct 
tensile  strength  of  the  concrete  itself.  This  statement  is  true,  of 


Fig.  38.     Table  of  Coefficients  of  Moments  over  the  Supports  Beams  Freely  Supported. 

course,  only  where  ample  lap  of  the  rods  has  been  provided  over 
the  support  which  should  always  be  attended  to  in  order  to  ensure 
safe  and  satisfactory  results  in  beam  design. 

Independent  beam  construction  has  been  used  to  a  small  extent, 
that  is  separate  beams  cast  in  the  shop  and  sent  to  the  job  as  individu- 
al units.  Difficulties  in  handling  made  up  units  in  erection  offset 
some  saving  in  cost  of  forming,  while  their  lack  of  joint  action  in 
carrying  load  puts  constructions  of  this  kind  at  a  serious  disadvantage 
compared  with  monolithic  work  wherever  the  loads  to  be  carried 
are  of  considerable  magnitude.  Independent  units,  however,  may 
be  quite  economically  employed  for  roof  construction  and  the  lightest 
kind  of  work  where  suitable  facilities  are  at  hand  for  carrying  on  the 
work  of  erection  economically. 


100 


MOMENTS    AND    REACTIONS    OF    CONTINUOUS    BEAMS 


For  convenient  reference,  Fig.  38  gives  the  bending  moment 
over  the  supports  in  a  series  of  equal  spans  uniformly  loaded  up  to 
nine  spans,  while  Fig.  39,  gives  the  maximum  moment  near  the 
mid  span  for  a  similar  series  of  continuous  beams  freely  supported 
and  uniformly  loaded.  From  these  moments,  taking  into  considera- 
tion the  relative  rigidity  of  the  column  and  the  beam  in  question, 
such  cases  as  those  above  suggested  where  the  rigidity  of  the  column 
is  small  in  comparison  to  the  beam,  may  be  correctly  treated.  The 
reactions  over  the  support  are  shown  in  Fig.  40.  The  amount  of 
bending  moment  and  the  reactions  are  to  be  determined,  of  course, 
by  multiplying  the  coefficients  in  the  table  by  W  the  total  load  per 


/' 


tt/AO.  077  X  0. 036  XO.  036  X  O. O77 


ISAO.O77XO.033X0.046  XO.O33  XO.O77 


/       \        /       \        /       \       / 

y^G044\(o.  o* /\(o.  O4/xa 
\    /        \    /        \    /        N 


Fig.  39.     Table  of  Coefficients,  Maximum   Positive   moment  between  Supports    for    Continuous 

Beam  Freely  Supported. 

span  by  the  length  of  span  to  obtain  the  bending  moment,  while 
the  reactions  are  the  tabulor  coefficients  of  W  the  total  load  on  a 
single  span. 

It  is  here  in  order  to  call  attention  to  the  fundamental  relation 
of  moment  magnitudes  which  may  be  observed  in  these  tables. 

For  any  span,  half  the  sum  of  the  moments  over  the  supports 
plus  the  moment  at  mid  span  is  a  constant,  equal  to  |  W  L,  in  which  W 
is  the  total  load  uniformly  applied  and  L  the  span.  This  relation 
is  of  use  in  treating  many  problems.  The  variation  of  the  maxi- 
mum positive  moment  from  the  moment  at  mid  span  is  greatest 
in  the  case  of  a  continuous  beam  of  two  spans  where  it  is  12^  per  cent. 
This  difference  is  much  less  in  all  other  cases  and  nothing  at  all 
whenever  the  moment  over  both  supports  is  of  equal  magnitude. 


ECONOMIC 


101 


It  will  be  noted  that  the  greatest  moment  at  mid  span  for  uni- 
form load  in  the  case  of  the  continuous  beam  of  two  spans  is  9  /1 28 
W  L,  and  the  maximum  positive  moment  for  a  span  of  indefinite 
length  is  WL  /24.  Further  it  will  be  noted  in  case  of  alternate 
spans  unloaded  that  the  usual  moment  provided  for  as  recommended 
for  live  load  is  1/16  W  L.  Hence  it  is  apparent  that  where  the 
columns  are  fairly  rigid  and  the  beams  are  integral  with  the  columns, 
little  attention  need  ordinarily  be  paid  as  to  whether  it  is  an  inter- 
mediate or  an  end  span  with  which  we  are  dealing. 

10.  Economic  Design  for  Beams.  As  already  noted,  the  co- 
efficient for  bending  of  a  continuous  beam  is  one-third  as  great 


Fig.  40.     Table  of    Reaction    Coefficients    for    Continuous    Beams    of    Equal    Spans    Uniformly 
Loaded  and  Freely  Supported. 


at  the  center  as  in  the  case  of  a  simple  beam,  and  two-thirds  at 
the  support.  Now  for  safety  ample  lap  of  the  bars  is  needed, 
hence  by  carrying  a  part  of  the  reinforcing  rods  required  at  the  center 
up  over  the  support  and  by  carrying  them  to  about  the  point 
of  contraflexure  or  so  far  that  the  negative  moment  in  case  of  a 
single  panel  load  will  be  taken  care  of  by  slab  reinforcement  parallel 
to  the  beam  we  have  need  theoretically  (considering  moment  only) 
two-thirds  the  section  of  steel  for  about  one-third  of  the  length 
and  one-third  of  the  section  of  metal  for  two-thirds  the  length  of 
this  beam  of  that  required  for  a  simple  beam.  In  other  words,  we 
have  the  following  comparison  from  the  standpoint  of  theoretical 
economy.  That  the  metal  required  for  a  continuous  beam  is  one- 
half  that  required  for  a  simple  beam  and  further  that  the  construe- 


102  ECONOMIC    STEEL    RATIO 

tion  with  a  continuous  beam  is  safer  to  erect  since  the  work  is  more 
securely  tied  together  and  it  can  be  depended  upon  with  a  good 
concrete  not  to  fail  suddenly  but  only  by  the  actual  stretching  out 
of  the  metal  to  the  point  of  ultimate  fracture  in  case  of  loading  equal 
to  three  or  four  times  that  which  it  was  calculated  to  sustain. 

This  theoretical  economy  however  cannot  be  fully  realized. 
Two-thirds  in  place  of  one-half  would  be  nearly  the  limit  attainable. 

Evidently  the  greater  the  depth  the  less  steel  will  be  required  to 
carry  a  given  load.  Usually,  however,  the  depth  to  be  used  in  an 
ordinary  building  is  determined  from  the  standpoint  of  appearance 
and  the  extra  cost  of  walls  for  a  given  clear  story  height  rather  than 
from  the  theoretically  economical  proportions  of  steel  and  concrete 
alone. 

A  mistake  which  is  frequently  made  is  in  building  beams  too 
narrow  and  deep  especially  where  they  are  spaced  closely.  Such 
construction  is  lacking  in  resistance  to  high  temperatures  since  too 
great  an  area  is  exposed  and  it  should  preferably  be  avoided  on  that 
account. 

A  minimum  width  of  ten  to  twelve  inches  should  be  adhered  to 
for  reinforced  concrete  beams  in  a  building  that  is  intended  to  be 
fireproof  to  a  high  degree,  and  such  a  width  for  moderate  spans  of 
sixteen  or  eighteen  feet  will  usually  give  ample  concrete  to  properly 
surround  the  reinforcement  in  the  beams. 

In  general  there  should  be  sufficient  width  to  allow  one  inch 
of  concrete  between  the  bars  or  a  width  not  less  than  one  and  one- 
fourth  times  the  diameter  of  the  bar  if  the  bars  are  parallel  for  any 
considerable  length.  Where  the  bond  shear  is  small  and  they  are 
bunched  as  at  the  top  of  the  beam  where  there  is  ample  spread  be- 
yond this  point  in  the  beam  this  requirement  becomes  of  no  especial 
importance. 

Relative  to  the  economic  proportion  of  concrete  and  steel,  the 
general  relation  to  be  observed,  is  that  the  amount  of  steel  decreases 
with  the  depth  of  the  beam,  while  the  amount  of  the  concrete  increases. 
With  one  percent  of  steel  as  the  element  of  reinforcement,  it  is  evident 
that  the  concrete  element  will  cost  more  than  the  steel  element 
on  the  basis  of  five  dollars  a  yard  for  the  concrete  and  steel  at 
fifty  dollars  per  ton.  Hence  with  ordinary  values  of  steel  and 
concrete,  the  limiting  permissible  percentages  and  relation  of  safe 
working  stresses  fix  economic  proportions.  In  the  continuous  beam, 
however,  where  there  is  double  reinforcement  over  the  support, 


FACTOR    OF    SAFETY  103 

a   nearer   approximation   of   the  balance   of   the   cost  of   concrete 
and  metal  may  be  approached  than  in  the  simple  beam. 

However,  in  the  T-beam  which  is  the  usual  construction  in 
buildings  this  percentage  would  be  based  upon  the  area  of  the  beam 
below  the  slab  plus  the  area  of  that  part  of  the  slab  above  and  oh 
each  side  of  the  rib  which  it  is  permissible  to  consider  as  forming  part 
of  the  compression  flange  of  the  beam,  and  the  economic  propor- 
tions would  have  a  smaller  proportion  of  steel,  since  the  portions  of 
the  slab  figured  in  with  the  beam  are  not  added  material  as  far  as 
the  beam  is  concerned,  and  hence  the  comparison  should  be  based 
more  properly  upon  the  area  of  the  rib  below  the  slab  of  the  concrete 
added  to  form  the  beam.  Hence  the  economic  proportion  of  steel 
would  in  general  be  reduced  below  that  of  the  limiting  proportion 
fixed  to  secure  conservative  working  stresses  in  the  concrete. 

This  conclusion,  that  the  cost  of  the  steel  in  the  T-beam  should 
be  less  than  the  concrete  is  strengthened  by  the  consideration  that  the 
cost  of  the  centering  increases  with  increase  of  depth  of  the  beam. 
These  practical  considerations  seem  to  have  been  entirely  over- 
looked in  the  discussion  of  the  paper  presented  to  the  American 
Society  of  Civil  Engineers  in  1906  by  Capt.  Sewell,  on  the  subject 
of  economic  construction  of  reinforced  concrete  floors. 

No  mathematical  formula  can  be  devised  which  will  take  into 
consideration  all  of  the  variable  elements  of  cost.  Trial  designs 
and  the  practical  judgment  of  the  constructor  enable  him  to  find 
an  approximate  and  satisfactory  solution  of  this  complex  problem. 
The  intimate  relation  of  horizontal  shear  to  permissible  percentage 
of  steel  which  we  have  pointed  out  earlier  in  the  discussion  is  but 
another  of  the  complex  elements  entering  into  the  problem. 

11.  Safe  Loads  for  and  Tests  of  Reinforced  Concrete  Construc= 
tion.    The  Joint  Committee  of  the  American  Society,  etc.,  in  their 
treatment  of  working  stresses  lay  down  this  commendable  rule : 

"In  selecting  the  permissible  working  stress  to  be  allowed  on 
concrete,  we  should  be  guided  by  the  working  stresses  usually 
allowed  for  other  materials  of  construction  so  that  all  structures 
of  the  same  class  but  composed  of  different  materials  may  have 
approximately  the  same  degree  of  safety." 

12.  True  and  Nominal  Factor  of  Safety.    A  popular  misconcep- 
tion regarding  the  meaning  of  the  term  factor  of  safety  as  applied 
to  steel  construction  has  exerted  an  influence  from  the  economic 
standpoint  adverse  to  the  rapid  introduction  of  concrete  construction. 

Many  have  the  mistaken  idea  that  the  factor  of  safety  of  four 


104  FACTOR    OF    SAFETY 

in  steel  construction  means  that  the  construction  may  be  safely 
loaded  to  four  times  the  rated  working  capacity;  but  this  is  not  the 
case,  since  the  yield  point  of  steel  is  only  about  twice  the  working 
load;  hence  the  actual  factory  of  safety  is  practically  two  against 
the  nominal  factor  of  four. 

In  other  words,  the  nominal  factor  of  safety  of  four  in  structural 
steel  work  is  based  on  the  ultimate  carrying  strength  in  tension  of 
the  metal  which  is  about  four  times  the  working  load,  but  after 
the  load  has  reached  a  little  more  than  double  the  working  load  the 
yield  point  value  of  the  steel  has  been  nearly  or  quite  reached  and  it 
commences  to  stretch,  pulling  out  in  case  of  mild  steel  before  break- 
ing sometimes  as  much  as  twenty  percent  or  more  of  its  total  length. 
Evidently  when  this  plastic  distortion  commences  in  a  beam  or 
column  the  member  is  soon  so  deformed  that  we  cannot  figure  its 
strength  in  the  frame,  thus  limiting  the  ultimate  strength  to  practical- 
ly a  little  more  than  twice  the  working  load  for  this  nominal  factor 
of  four. 

In  properly  designed  concrete  construction  the  concrete  is  made 
stronger  than  the  steel,  for  one  reason  because  it  is  generally  econom- 
ical so  to  do,  and  hence  the  strength  of  the  steel  is  the  strength  of 
the  reinforced  concrete  construction  and  it  would  not  be  reasonable 
to  expect  to  subject  the  steel  to  higher  stresses  in  the  case  of  concrete 
construction  than  is  permissible  in  structural  work;  hence  twice 
the  wrorking  load  is  a  fair  test  for  this  type  of  work.  In  reality,  in 
view  of  the  fact  that  the  cement  improves  with  age,  if  it  will  stand 
this  test  at  an  age  of  from  three  to  four  months  the  owner  can  rest 
assured  that  the  factor  of  safety  is  greater  than  with  structural 
steel  construction. 

Referring  to  the  specifications  for  reinforcing  bars,  page  (34), 
it  will  be  noted  that  for  structural  grade  bars  (recommended  for 
beams  and  bent  work)  the  yield  point  is  33,000,  or  two  and  one- 
fifth  times  the  working  stress  of  16,000  pounds  allowed  by  nearly 
all  building  codes,  while  for  hard  grade,  50,000  pounds  per  square 
inch  is  the  yield  point  value.  Accordingly,  higher  test  loads  can 
be  applied  where  the  reinforcement  is  of  hard  grade  steel  than  with 
the  softer  grade.  However,  greater  care  is  necessary  in  bending 
the  hard  grade  metal;  the  structure  is  not  so  tough  and  the  results 
of  the  use  of  this  grade  of  steel  are  more  uncertain. 

Excessive  tests  are  not  to  be  recommended,  since  some  permanent 
set  and  weakening  of  the  structure  may  result  therefrom.  Elastic 


TEST    LOADING 


105 


deportment  in  accord  with  theory  under  tests  of  one  and  three  quar- 
ters to  two  times  the  working  stress  for  heavy  work  should  suffice. 

13.  Method  of  Loading  for  Tests.  To  secure  results  of  scientific 
value  the  material  used  for  test  loads  should  be  piled  in  such  a 
manner  that  its  action  on  the  slab  or  beam  under  consideration  shall 
not  be  masked  by  arch  action. 

A  misleading  type  of  test  is  shown  in  Fig.  41,  consisting  of  cast 
iron  piled  up  in  a  manner  which  enables  it  to  arch  readily  to  a  large 


Fig.  41. 


Test  in  which  Arch  Action  occurs  from  Main  Beam  to  Main  Beam,  giving 
a  Misleading  indication  as  to  the  Strength  of  the  Floor. 


extent  from  main  beam  to  main  beam.  In  this  case  the  construction 
is  practically  type  I,  with  the  joist  girders  five  feet  to  six  feet  apart. 
The  load  shown  was  actually  1,500  pounds  per  foot,  but  so  far  as 
the  girder  on  which  it  rested  was  concerned  it  was  probably  not 
equivalent  to  more  than  1,000  pounds  uniform  load  placed  in  a 
manner  which  would  prevent  arch  action  from  main  girder  to  main 
girder. 

A  material  such  as  gravel  in  bulk  may  arch  somswhat,  perhaps 
to  the  extent  of  five  to  six  percent.  With  cement  sacks  there  may 
be  also  a  small  amount  of  arch  action,  but  in  view  of  the  fact  that 


106 


ARCH    ACTION    IN    TESTS 


the  material  is  not  rigid  in  form,  as  in  the  case  of  pig  iron,  this  action 
can  amount  to  very  little  unless  special  pains  be  taken  to  lay  the 
bags  in  a  manner  to  secure  such  action,  and  even  with  the  greatest 
pains  it  is  doubtful  whether  the  bags  can  be  placed  on  a  large  panel 
in  such  a  manner  as  would  make  the  arch  action  amount  to  more  than 
twice  the  above  limits. 

In  considering  the  degree  or  amount  of  arch  action  which  may 
exist  in  a  pile  of  material  it  may  be  noted  in  first  place  that  the  arch 
action  will  be  greater  the  greater  the  height  of  the  pile  as  compared 
with  its  base. 


Fig  42.     Test  in  which  Action  is  Eliminated  using  Pig  Iron. 

Thus  a  pile  of  gravel  seventeen  feet  square  held  in  by  a  wall 
of  sacks  filled  with  gravel  on  each  side  eight  feet  high  might  reduce 
the  actual  bending  on  the  slab  five  to  eight  percent.  If  the  pile 
were  one-third  of  this  height  probably  the  arch  action  would  not 
exceed  one-half  to  three-quarters  of  one  percent. 

With  a  pile  of  pig  iron  carefully  built  the  amount  of  arch  action 
might  readily  become  large,  since  the  pigs  are  rigid,  and  if  laid  up 
carefully  a  quite  perfect  Hindu  arch  could  readily  be  built  which 
would  carry  over  half  the  load  to  the  support  or  main  beam  without 
straining  the  girder  or  slab  which  it  is  nominally  the  intent  to  test. 

Fig.  42  is  a  test  made  at  the  St.  Louis  Exposition,  using  cast 
iron,  in  which  there  can  be  no  doubt  as  to  the  distribution  of  the 
load. 


LOAD  AREA  FOR  SATISFACTORY  TESTS  107 

In  general,  the  contractor  desires  to  use  for  loading  the  materials 
about  the  work  which  can  be  conveniently  placed  upon  the  panel 
or  area  to  be  tested  and  he  should,  of  course,  be  allowed  to  do  this, 
since  the  expense  of  making  a  reasonably  conclusive  test  on  a  floor 
will  frequently  amount  to  several  hundred  dollars. 

Brick,  cement  in  sacks,  sand  or  gravel,  stone,  plaster,  and  the 
like  will  frequently  be  used  by  the  contractor  if  he  has  them  at 
hand,  instead  of  carting  in  pig  iron  from  a  distance,  unless  there  is 
some  object  to  be  gained  from  a  misleading  test. 

The  area  necessary  to  be  covered  in  making  a  satisfactory  test 
of  a  building,  will,  of  course,  depend  on  the  type  of  construction 
and  the  unit  which  it  is  desired  to  investigate.  In  the  case  of  the 
slab  between  parallel  beams,  Type  II,  the  loaded  length  parallel 
to  the  beams,  should  be  not  less  than  two  times  the  distance 
between  the  beams  in  order  to  induce  a  condition  of  maximum  stress 
at  the  center  of  the  loaded  area  approximately  equal  to  that  which 
would  occur  if  the  full  area  of  the  slab  were  loaded. 

In  the  case  of  continuous  slabs,  Types  I  and  II,  the  most  severe 
positive  stress  is  determined  by  loading  placed  upon  single  panels 
or  alternate  panels. 

In  the  Mushroom  construction,  the  maximum  deflection  at  the 
diagonal  center  of  the  panel  is  secured  by  loading  one  panel  only. 
The  maximum  possible  stress  over  the  column  for  a  given  unit 
intensity  of  load  occurs  when  four  panels  are  loaded,  tho  this  being 
a  uniform  stress  all  around  the  column  it  frequently  is  not  in  excess 
of  the  unit  stress  in  compression  at  the  underside  of  the  cap  with 
the  unbalanced  load  of  the  single  panel  loaded,  and  as  the  steel  is 
usually  in  excess  the  test  of  four  panels  leads  to  no  more  know- 
ledge of  the  deportment  of  the  structure  than  would  be  obtained 
by  the  single  panel  test. 

In  a  flat  slab  on  spaced  supports,  reinforced  in  two  directions, 
the  maximum  deflection  at  the  diagonal  center  of  the  panel  is  obtained 
when  five  panels  are  loaded,  the  panel  under  consideration  and  the 
four  panels  adjacent  to  its  sides.  This  difference  in  deportment  is 
brought  about  by  the  fact  that  the  two  way  reinforced  slab  throws 
the  shear  on  the  side  belts  whereas  the  four  way  reinforcement  tends 
to  transfer  it  more  directly  to  the  column  center.  In  other  words, 
while  the  mode  of  operation  of  the  two  types  is  substantially  the 
same  as  regards  cantilever  head  and  the  character  of  the  stress  about 
the  diagonal  center  of  the  panel,  it  is  otherwise  with  the  distribution 


108 


SHEAR    IX    BEAMS 


of  shear  in  cross  sections  of  the  direct  belts  which  act  after  the  man- 
ner of  beams  in  both  constructions,  the  two  way  reinforcement  not 
taking  advantage  of  all  of  the  advantageous  characteristics  of  the 
four  way  system. 

14.  Shears  in  Beams.  Shearing  stress  at  and  near  the  supports 
of  a  cantilever  or  continuous  beam  or  slab  is  an  action  of  an  essen- 
tially different  kind  from  the  shear  accompanying  bending  in  a  simple 
beam.  At  the  support  of  a  uniformly  loaded  continuous  beam  for 
example,  where  the  negative  moment  reaches  its  greatest  numerical 
value,  the  beam  resists  a  sliding  stress  on  its  vertical  cross  section 
equal  to  the  load  transmitted  by  the  beam  to  the  support.  This  is 
accompanied  by  no  horizontal  shearing  stress  across  this  section, 
and  no  diagonal  tensional  stress  is  called  into  play  by  this  sliding 
shear,  which  may  be  otherwise  designated  as  punching  shear,  never- 
theless diagonal  shearing  deformation  occurs  here  as  will  be  shown 
later. 

We  will  now  consider  how  it  may  be  true  that  there  is  no  hori- 
zontal shearing  stress  in  this  case,  a  conclusion  which  is  entirely 
opposed  to  the  principles  underlying  ordinary  bending  shear  where 
statical  equilibrium  requires  the  intensity  of  shear  on  vertical  and 
horizontal  planes  to  be  equal  at  all  points  of  the  material. 

In  Fig  A  let  A  B  represent  a 

£  &'  vertical  section  at  the  edge  A  of 

the  support  of  a  continuous  beam 
at  the  left  of  A,  while  A'Bf  is  a 
neighboring    vertical    section    of 
ihe  beam.     This  element  of 
length  of  the  beam  is  subjected 
to    unequal    tensile    stresses    on 
those  points  of  its  vertical  faces 
lying  above  the  neutral  axis  N 
and  to  unequal  compressive 
stresses  on  the  faces  below  N. 

Let  T  represent  the  difference 
of  the  tensile  stresses  and  C  =  —T 

the  difference  of  the  compressive  stresses  on  the  opposite  faces. 
This  difference  is  greater  per  unit  of  length  at  the  support  than  else- 
where, as  appears  from  the  greater  slope  of  the  moment  curve  here. 
These  differences  or  resultant  horizontal  forces  on  the  faces  form  a 
couple  which  acts  on  the  element  ABB' A'.  This  couple  is  held  in 
equilibrium  by  the  couple  arising  from  the  vertical  shearing  stresses 


/v 

,5 

c  — 

A           A' 

Fig.  A 

SHEAR    IN    BEAMS  109 

on  the  opposite  faces  AB  and  A'B' ' .  The  stresses  T  and  C  do  not, 
however,  cause  shears  between  the  horizontal  fibers  but  merely 
cause  differences  between  the  tensions  or  compressions  at  their 
extremities  which  determine  the  law  of  distribution  of  the  intensity 
of  the  total  vertical  shear  S  on  AB  and  A'B'  and  make  it  increase 
as  T  and  C  do,  viz:  proportionally  to  distance  from  the  neutral 
axis  N. 

So  long  as  the  reinforcement  at  the  top  or  tension  side  of  the  beam 
or  slab  at  the  support  preserves  the  concrete  perfectly  intact  it  will 
compel  the  concrete  to  act  in  the  manner  just  indicated.  We  shall 
designate  this  action  as  punching  shear  altho  it  does  not  conform  to 
the  description  of  punching  shear  as  used  by  the  Joint  Committee, 
since  they  do  not  allow  any  compressions  upon  AB  or  A'B' .  It  is 
doubtful  whether  such  a  state  of  stress  is  possible  as  that  described 
in  the  definition  of  the  Joint  Committee. 

The    diagonal    tensional    deformations    of    punching    shear    are 

not  the  same  as  in  ordinary 
bending  shear  as  may  be  seen 
from  the  accompanying  repre- 
sentation, Fig.  B.  In  this  dia- 
gram if  a  vertical  shearing  stress 
of  given  intensity  on  A  B  will 
cause  DC  situated  at  a  distance 
of  one  unit  from  AB  to  be  dis- 
placed to  Z)'C",  then  in  homo- 
geneous material  an  equal  hori- 
zontal shear  such  as  occurs  in 
bending  shear  will  displace  AC 
Fig  B  an  equal  amount,  so  that  the 

total  diagonal  displacement  CC" 

=  CC'  V  2  is  half  of  it  due  to  each.  Consequently  a  punching  shear 
in  homogeneous  materials  not  accompanied  by  horizontal  shearing 
stress,  causes  a  diagonal  deformation  only  one-half  as  great  as  is 
caused  by  bending  shear  where  the  intensity  on  the  horizontal  plane 
is  equal  to  that  on  the  vertical  plane.  Hence  only  half  as  much 
diagonal  reinforcement  would  be  needed  to  restrain  diagonal  elon- 
gation in  the  one  case  as  would  be  required  in  the  other.  If  the 
mean  unit  resistance  to  horizontal  shear,  however,  is  less  than  to 
vertical  shear,  the  horizontal  and  vertical  deformations  CC'  and 
C'C"  will  be  unequal,  as  well  as  their  diagonal  components  CE  and 
EC".  But  they  may  be  readily  computed  when  the  mean  moduli 
of  vertical  and  horizontal  resistance  are  known. 


110  PUNCHING    SHEAR 

Experimental  determinations  of  the  strength  of  concrete  in  resis- 
tance to  direct  sliding  shear  not  involving  diagonal  tension  show  that 
in  general  it  exceeds  50  percent  of  its  compressive  strength  but  its 
strength  is  largely  dependent  upon  the  age  of  the  concrete,  especially 
in  resistance  to  punching  shear  at  supports. 

Taylor  and  Thompson*  quote  Spofford's  Experiments  on  sliding 
shear  in  concrete  from  24  to  32  days  old  and  say  "  these  experiments 
gave  a  shearing  strength  ranging  from  60  to  80  percent  of  the  com- 
pressive strength  of  the  concrete,  which  agrees  substantially  with 
the  experiments  of  Prof.  Arthur  N.  Talbot  in  1906." 

In  order  that  any  such  values  of  direct  shear  should  exist  in  a 
beam  or  slab  the  anchorage  of  the  tension  steel  must  be  absolutely 
secure  and  its  amount  sufficient.  In  this  lies  the  difference  between 
continuous  beams  and  slabs  where  such  anchorage  exists,  and  footings 
where  anchorage  is  relatively  insecure. 

As  stated  previously,  punching  shear,  which  is  the  shear  at  or 
near  supports  of  continuous  beams,  is  distributed  on  the  vertical 
section  in  such  a  manner  as  to  be  greatest  at  the  extreme  fiber  and 
it  is  entirely  unsafe  to  trust  to  the  stability  of  unreinforced  concrete 
to  resist  it.  Steel  rods  should  always  be  put  in  both  the  top  and 
bottom  layers  of  continuous  beams,  more  in  the  top  than  in  the 
bottom  and  continued  entirely  across  the  top  of  the  support  and  well 
anchored  at  some  distance  into  the  next  span.  The  direct  shear  of 
such  steel  like  that  of  rivets  can  be  counted  on  with  certainty,  as 
concrete  cannot. 

With  a  1  :  2  :  4  mix,  28  days  old,  cured  under  laboratory  condi- 
tions and  containing  .75  of  1  percent  of  reinforcement,  the  section 
may  be  counted  on  to  resist  safely  a  shearing  stress  of  6  percent  of 
the  compressive  strength  of  the  concrete,  and  with  over  2  percent 
of  tensile  steel  and  two-thirds  as  much  in  compression,  this  may  be 
increased  30  to  50  percent.  But  no  28  day  concrete  should  be  sub- 
jected to  severe  bending  and  shearing  stress  if  it  can  be  avoided. 
Resistance  to  combined  bending  and  shearing  develops  much  more 
slowly  and  much  later  than  to  compression.  But  when  a  beam  has 
been  well  cured  for  90  days  in  good  drying  weather,  the  shearing 
strength  at  the  supports  is  much  more  than  double  that  just  stated 
for  28  day  concrete. 

Under  the  conditions  which  have  been  outlined  it  is  evident  that 
the  integrity  of  the  section  is  primarily  dependent  on  the  sufficiency 


'Concrete  Plain  and  Reinforced,  2nd  Ed.,  p.  382. 


COMPUTING    SHEAR    RESISTANCE 


111 


and  proper  distribution  of  the  reinforcing  steel.  It  is  necessary 
therefore  to  introduce  the  reinforcement  as  a  principal  element  of 
strength  into  the  computation  of  the  safe  vertical  resistance  to  shear 
at  the  supports  of  continuous  beams,  which  has  been  done  as  follows : 

For  reinforcement  arranged  as  shown  in  Figs.  35,  36  and  37,  so 
that  it  has  ample  anchorage  on  each  side  of  the  support,  when  the 
working  stress  in  the  concrete  is  assumed  to  be  merely  its  resistance 
to  diagonal  tension  of  40  Ibs.  per  square  inch  for  a  1  :  3  :  5  mix,  of 
50  Ibs.  for  a  1  :  2  : 4  mix,  or  of  65  Ibs.  for  a  1  :  1  \  :  3  mix,  the  steel 
in  the  upper  flange  may  be  safely  counted  on  for  a  working  stress 
in  shear  of  10,000  Ibs.  per  square  inch,  and  that  in  the  bottom  for 
half  as  much. 

This  method  is  illustrated  in  the  following  computations  of  the 
allowable  working  stresses  in  the  continuous  beam  in  Fig.  21,  illus- 
trating the  Minneapolis  Paper  Company  building,  tested  after  the 
concrete  was  well  cured  for  more  than  90  days: 


Area  of 

Section  in 
Inches 

Square 

Stress  Ibs.  per 
Square  Inch 

Working  Resis- 
tance in  Lbs. 

Concrete  . 

.  .  .  240 

50 

12000 

Top  steel  6  rods.  .  .  . 
Bottom  steel  4  rods. 

.  .  .     6 
.  .     2.4 

10000 
5000 

60000 
12000 

Stirrups  4 

rods  

.44 

15000 

6600 

90600 

With  a  design  unit  load  of  500  Ibs.  upon  this  floor,  the  load  may 
be  assumed  to  have  been  carried  by  the  beams  nearly  in  proportion  to 
their  lengths  and  then  the  total  load  upon  one  of  the  longer  beams 
would  be 

15.33X21.5X500X21.5 /(21.5  +  15.33)  =92.450  Ibs. 
and  the  shear  at  a  support  one  half  this,  or  46,225  Ibs.  The  beam 
was  tested  to  nearly  double  this  amount,  or  92,450  Ibs.,  which  agrees 
with  the  working  stress  previously  computed.  But  this  according 
to  the  above  computations  was  not  half  what  the  beam  would  have 
carried  safely. 

It  will  appear  from  this  investigation  that  the  critical  section 
for  shear  in  a  beam  is  not  at  the  support  in  a  continuous  beam, 
neither  is  it  at  the  support  of  a  properly  designed  simple  beam  with 
steel  carried  past  the  supports  both  at  the  top  and  bottom  and 


112  JOINT  COMMITTEE'S  RECOMMENDATIONS 

properly  anchored.  Failure  from  diagonal  shear  will  occur  in  the 
continuous  beam  nearer  the  points  of  inflection  and  in  the  simple 
beam  a  little  way  from  the  supports,  dependent  upon  the  arrange- 
ment of  the  sloping  reinforcing  rods  and  stirrups. 

In  the  construction  of  the  simple  beam  type,  the  recommendations 
of  the  Joint  Committee  quoted  herewith  are  conservative,  tho 
these  rules  cannot  be  reasonably  applied  in  determining  the  shearing 
resistance  of  scientifically  designed  continuous  beams  such  as  those 
outlined: 

15.  Shear  and  Diagonal  Tension. — "In  calculations  on  beams 
in  which  the  maximum  shearing  stress  in  a  section  is  used  as  the 
means   of   measuring   the   resistance   to   diagonal  tension   stress, 
the  following  allowable  values  for  the  maximum  vertical  shearing 
stress  are  recommended: 

(a)  For  beams  with  horizontal  bars  only  and  without  web  rein- 
forcement  calculated  by   Formula    (22):   2%   of  the   compressive 
strength,     (i.  e.  for  bottom  reinforcement  only.) 

(b)  For  beams  thoroly   reinforced  with  web   reinforcement: 
the  value  of  the  shearing  stress  calculated  as  for  (a),  (that  is,  using 
the  total  external  vertical  shear  in  Formula  (22)  for  shearing  unit- 
stress),  must  not  exceed  6%  of  the  compressive  strength.     The  web 
reinforcement,   exclusive  of  bent-up  bars,   in  this  case,   shall  be 
proportioned  to  resist  two-thirds  of  the   external   vertical  shear   in 
Formulas  (24)  or  (25).. 

(c)  For  beams  in  which  part  of  the  longitudinal  reinforcement 
is  used  in  the  form  of  bent-up  bars  distributed  over  a  portion  of 
the  beam  in  a  way  covering  the  requirements  of  this  type  of  web 
reinforcement:  the  limit  of  maximum  vertical  shearing  stress  (the 
stress  calculated  as  for  (a)  ),  3%  of  the  compressive  strength. 

(d)  Where  punching  shear  occurs,  that  is,  shearing  stress  un- 
combined  with  compression  normal  to  the  shearing  surface,  and 
with  all  tension  normal  to  the  shearing  plane  provided  for  by 
reinforcement:  a  shearing  stress  of  6%  of  the  compressive  strength 
may  be  allowed." 

But  since  we  are  of  the  opinion  that  these  recommendations  are 
not  applicable  to  columns,  working  stresses  for  columns  will  be 
specially  treated  under  that  heading.  The  committee  failed  to 
recognize  the  action  of  bond  shear  in  column  and  multiple  way 
slab  construction,  which  action  is  here  regarded  as  an  essential 
factor  in  assigning  their  working  stresses.  Such  recognition  was 
perhaps  not  to  be  expected  in  view  of  the  fact  that  this  subject  has 
not  heretofore  been  adequately  treated  in  the  literature  of  reinforced 
concrete. 

16.  Working  Stresses — General  Assumptions:  The  following 
working  stresses  are  recommended  for  static  loads.     Proper  allow- 
ances for  vibration  and  impact  are  to  be  added  to  live  loads  where 
necessary  to  produce  an  equivalent  static  load  before  applying 
the  unit  stresses  in  proportioning  parts. 

In  selecting  the  permissible  working  stress  to  be  allowed  on 
concrete,  we  should  be  guided  by  the  working  stresses  usually 
allowed  for  other  materials  of  construction,  so  that  all  structures 


WORKING    STRESSES  113 

of  the  same  class  but  composed  of  different  materials  may  have 
approximately  the  same  degree  of  safety. 

The  following  recommendations  as  to  allowable  stresses  are 
given  in  the  form  of  percentages  of  the  ultimate  strength  of  the 
particular  concrete  which  is  to  be  used;  this  ultimate  strength  is 
to  be  that  developed  in  cylinders  8  in.  in  diam  ;ter  and  16  in.  long, 
made  and  stored  under  laboratory  conditions,  at  an  age  of  28  days. 
In  the  absence  of  definite  knowledge,  in  advance  of  construction,  as 
to  just  what  strength  may  be  expected,  the  Committee  submits  the 
following  values  as  those  which  should  be  obtained  with  materials 
and  workmanship  in  accordance  with  the  recommendations  of  this 
report. 

Although  occasional  tests  may  show  higher  results  than  those 
here  given,  the  Committee  recommends  that  these  values  should  be 
the  maximum  used  in  design. 

TABLE    OF    STRENGTHS    OF    DIFFERENT    MIXTURES 
OF  CONCRETE 

(In  pounds  per  square  inch) 

Aggregate                          1:1:2    1:1±:3  1:2:4  1:2^:5  1:3:6 

Granite,  trap  rock 3300       2800  2200  1800  1400 

Gravel,    hard    limestone    and 

hard  sandstone 3000       2500  2000  1600  1300 

Soft  limestone  and  sandstone2200       1800  1500  1200  1000 

Cinders 800        700  600  500  400 

Bearing:  When  compression  is  applied  to  a  surface  of  con- 
crete of  at  least  twice  the  loaded  area,  a  stress  of  32.5%  of  the 
compressive  strength  may  be  allowed. 

Axial  Compression:  For  concentric  compression  on  a  plain 
concrete  column  or  pier,  the  length  of  which  does  not  exceed  12 
diameters,  22.5%  of  the  compressive  strength  may  be  allowed. 

Compression  in  Extreme  Fiber:  The  extreme-fiber  stress  of 
a  beam,  calculated  on  the  assumption  of  a  constant  modulus  of 
elasticity  for  concrete  under  working  stresses,  may  be  allowed 
to  reach  32.5%  of  the  compressive  strength.  Adjacent  to  the 
support  of  continuous  beams  stresses  15%  higher  may  be  used. 

Bond:  The  bond  stress  between  concrete  and  plain  reinforcing 
bars  may  be  assumed  at  4%  of  the  compressive  strength,  or  2% 
in  the  case  of  drawn  wire. 

Reinforcement:  The  tensile  or  compressive  strength  in  steel 
should  not  exceed  16,000  Ib.  per  sq.  in. 

In  structural-steel  members,  the  working  stresses  adopted 
by  the  American  Railway  Engineering  Association  are  recom- 
mended." 

Under  the  heading  of  working  stresses  the  report  of  the  Joint 
Committee  deals  only  with  permissible  values  for  stresses  in  one 
direction.  Now  in  concrete  work  constructed  as  a  continuous 
monolith  the  material  is  frequently  strained  in  multiple  directions — 
for  example  in  Type  IV  floor  construction  the  bottom  portion  of 
the  continuous  slab  near  the  column  is  under  compression  radially 
toward  the  column  and  circumferentially  about  the  column. 

Morley  in  his  excellent  work  on  Strength  of  Materials  has  dis- 
cussed the  question  of  compound  stress  very  fully.  He  shows  that 
failure  in  elastic  materials  under  stress  results  not  from  balanced 
hydraulic  stresses  but  from  the  unbalanced  shearing  stresses. 


114  COMPOUND    TENSILE    STRESS 

Considere  in  a  valuable  series  of  tests  of  the  compressive  strength 
of  concrete  cylinders  found  that  the  endwise  compressive  resistance 
might  be  almost  indefinitely  increased  by  increase  of  lateral  hydraulic 
pressure.  These  tests  were  carried  to  the  extent  of  increasing  the 
crushing  resistance  of  cylinders  endwise  four  to  five  fold.  This 
increase  appeared  to  be  limited  only  by  the  amount  of  hydraulic 
pressure  applied  laterally. 

On  this  principle  the  safe  radial  compression  in  the  lower  part 
of  a  continuous  flat  slab  may  be  very  conservatively  taken  at  values 
double  those  for  direct  axial  compression  provided  suitable  provision 
is  made  for  shear. 

This  view  is  borne  out  by  practical  experience  with  thousands 
of  such  cases  in  which  no  evidence  of  weakness  has  been  observed 
with  good  concrete  thoroly  cured. 

Shear  and  tension  failures  are,  however,  more  liable  to  occur 
on  the  under  side  of  the  slab  near  the  cap  than  elsewhere  when  the 
cement  is  partly  cured  and  the  forms  have  been  prematurely  removed. 
In  this  case  the  inspector's  duty  is  to  first  investigate  this  zone  for 
soundness  and  remove  and  recast  any  damaged  material. 

As  in  the  case  of  beams,  full  advantage  of  the  maximum  com- 
pressive bending  resistance  can  be  taken  to  the  limit  only  of  a  certain 
ratio  of  thickness  to  span  and  proper  reduction  made  for  smaller 
ratios  as  discussed  in  the  treatment  of  permissible  steel  ratios  and 
shearing  stresses  for  this  type  of  construction. 

17.  Compound  Tensile  Strength.  The  same  reasoning  applies 
to  tensile  stress  that  applies  to  compressive  working  stress  when 
the  steel  is  distributed  in  the  form  of  small  rods  closely  spaced. 
One  of  the  facts  in  favor  of  multiple-way  reinforcement  in  the  natural 
concrete  types  is  that  the  direct  tensile  reistance  of  the  concrete 
is  increased  somewhat  by  strain  in  multiple  directions.  But  in  view 
of  the  fact  that  the  direct  tensile  resistance  of  concrete  is  only  one 
tenth  or  one  twelfth  its  compressive  resistance,  an  addition  of  forty 
to  fifty  percent  to  this  direct  tensile  resistance  of  concrete  does  not 
render  its  dependable  value  of  sufficient  magnitude  to  be  worthy 
of  consideration  as  a  safe  practical  element  of  strength,  and  as  in  the 
case  of  beams  it  should  be  disregarded  for  this  reason. 

Further  the  coefficient  of  expansion  or  contraction  being  .0000065, 
it  is  obvious  that  a  drop  of  temperature  of  25  degrees  will  overcome 
ordinary  direct  tensile  reistance  of  concrete  assuming  both  ends  to 
be  rigidly  restrained,  and  as  concrete  work  in  Northern  latitudes, 


CONCRETE    BEAM    AS    A    MECHANISM  115 

at  least,  is  frequently  subjected  to  a  range  of  temperature  much 
greater  than  this  below  the  temperature  at  which  the  concrete  has 
hardened,  we  are  not  justified  in  considering  direct  tensile  resistance 
as  a  dependable  element  of  strength  even  under  the  more  favorable 
condition  under  discussion. 

It  is  a  favorable  condition  in  building  construction  that  there 
is  more  or  less  chance  for  adjustment  of  moderate  temperature 
effects  and  that  columns  give  and  bend  in  and  out  by  small  amounts 
thus  accommodating  expansion  and  contraction  of  flooring,  and  the 
same  action  occurs  with  walls,  etc.,  otherwise  the  combination  in 
the  same  structure  of  different  materials  such  as  stone,  brick,  steel, 
terra  cotta,  etc.,  having  widely  different  coefficients  of  expansion 
in  the  same  building  would  not  give  satisfactory  results. 

18.     The   Reinforced    Concrete   Beam   as   a  Mechanism.     The 

combination  of  the  two  elements,  the  concrete  and  the  steel  in  the 
beam  constitutes  a  device  consisting  of  two  relatively  constrained 
parts  which  by  certain  predetermined  intermotions  serve  to  trans- 
mit force  and  motion  in  such  a  manner  as  to  produce  the  effect  of 
carrying  the  load  to  the  respective  supports  while  the  arrangement 
of  the  metal  in  its  position  vertically  and  horizontally  with  reference 
to  the  supports  determines  the  general  law  of  operation  of  the  device. 
This  operation,  nevertheless  must  conform  to  certain  fixed  or  fun- 
damental natural  laws.  These  fundamental  laws  form  the  basis 
of  the  theory  of  work  which  is  well  understood  and  generally  applied 
by  the  engineering  profession  in  the  treatment  of  bridge  and  frame 
structures  but  which  seems  to  have  been  to  some  extent  ignored 
in  case  of  such  a  mechanical  device  as  a  concrete  beam  or  floor. 

The  fundamental  laws  upon  which  the  theory  of  work  is  based 
are  derived  primarily  from  the  general  principle  known  as  the  law 
of  conservation  of  energy.  This  law  is  expressed  in  Merriman's 
Civil  Engineers'  Pocket  Book,  in  the  following  statement: 

"If  the  system  of  bodies  neither  receives  nor  gives  out  energy, 
then  its  total  store  of  energy,  all  forms  included,  remains  con- 
stant. There  may  be  a  transfer  of  energy  from  one  part  of  the 
system  to  the  other,  but  the  total  gain  or  loss  in  one  part  is  exactly 
equivalent  to  the  loss  or  gain  in  the  remainder." 

We  may  also  state  the  law  in  a  general  way  as  follows:  Energy 
can  be  transformed  or  changed  in  form  but  it  cannot  be  destroyed. 

When  we  -load  a  floor,  we  have  an  arrangement  or  device  by 
which  the  load  placed  on  the  floor  is  gradually  lowered  from  its 
original  position  to  the  lower  position  assumed  by  the  slab  as  it 


116  APPLYING    THE    METHOD    OF    WORK 

bends,  and  if  the  slab  is  elastic  the  actual  mechanical  energy  developed 
by  the  downward  motion  of  the  load  under  the  law  stated  must  be 
stored  as  potential  energy  of  elastic  deformation  within  the  substance 
of  the  floor.  This  direct  relation  is  ordinarily  expressed  in  the 
statement  that  the  external  work  of  the  load  is  transformed  into  in- 
ternal work  of  deformation.  This  principle  is  worked  out  in  great 
detail  in  designing  bridge  structures  to  determine  deflections  by  work 
done,  and  is  of  the  utmost  value  to  the  engineer  in  its  various  appli- 
cations. 

The  above  relation  was  expressed  in  1866  in  the  theorem  of 
Clapeyron,  (See  Lame,  "Lecons  sur  la  theorie  mathematique  de 
1'elasticite  des  corps  solides,"  deuxieme  edition  Paris,  1866),  and  is 
stated  as  follows: 

"The  exterior  force  applied,  multiplied  by  the  displacement 
in  the  direction  of  its  point  of  application,  equals  the  sum  of  all 
the  internal  work  of  a  body  elastically  deformed." 

This  theorem  is  a  direct  corollary  of  the  fundamental  law  of 
conservation  of  energy. 

In  applying  this  method  of  work,  we  of  course  consider  only  the 
elastic  deformation  of  the  beam  relative  to  its  points  of  support. 

When  a  newly  cured  beam  is  first  loaded  the  deflections  up  to 
a  point  where  the  reinforcement  is  strained  to  as  much  as  four  or  five 
thousand  pounds  per  square  inch,  are  about  half  or  less' than  half  of 
the  corresponding  deflection  for  the  same  increment  of  load  where 
the  steel  is  stressed  from  ten  to  twelve  thousand  pounds  per  square 
inch.  This  difference  in  measured  steel  stress  and  deflection  under 
the  laws  noted  indicates  a  difference  in  the  mode  of  operation  of  the 
mechanical  device  with  which  we  are  dealing  which  it  is  now  in  order 
to  investigate.  As  the  stress  in  the  steel  approaches  four  to  six  thous- 
and pounds  per  square  inch  the  direct  tension  in  the  concrete  cor- 
responding to  the  stress  in  the  steel  ranges  upwards  of  250  or  360 
pounds  per  square  inch,  or  approximately  the  ultimate  direct  tensile 
strength  of  the  concrete  matrix.  As  the  load  is  increased  further, 
a  yielding  of  the  concrete  takes  place  and  cracks  begin  to  develop. 
This  yielding  of  the  concrete  results  in  a  dissipation  or  loss  of  the 
mechanical  energy  stored  by  the  elastic  deformation  of  the  concrete 
and  new  energy  is  developed  by  the  motion  of  the  load  thru  increased 
deflection  until  equilibrium  is  re-established,  which  energy  is  stored 
up  in  a  stable  manner  in  the  steel. 

The  action  which  we  have  outlined  does  not  constitute  a  transfer 


STIRRUPS  H7 

of  the  stress  in  the  concrete  to  the  steel  as  some  incorrectly  may 
consider  it,  but  it  represents  the  dissipation  of  energy  thru  the  over- 
strain of  the  concrete,  the  development  of  new  energy  by  the  motion 
of  the  load  thru  increased  deflection  and  the  storage  of  this  new 
energy  in  the  steel  so  that  when  the  load  is  removed  the  internal 
work  then  stored  in  the  beam  is  only  that  stored  by  elastic  deforma- 
tion which  may  be  given  back  as  the  load  is  removed.  For  perfectly 
elastic  deformation  then  we  may  say  that  thru  indirect  stress  there 
is  an  amount  of  energy  stored  by  lines  of  indirect  tension  and  com- 
pression emanating  from  bond  shear  at  the  surface  of  the  steel  equal 
to  that  stored  in  the  steel  and  that  the  lines  of  principal  stress  in  the 
reinforced  beam  are  for  this  reason  somewhat  analogous  to  those  of 
a  beam  of  homogeneous  material.  When,  however,  energy  is  lost 
or  dissipated  thru  inelastic  stretching  or  cracking  of  the  concrete, 
a  different  distribution  of  internal  stress  results. 

The  indirect  stresses  in  tension  are  greatly  decreased  in  this  case 
while  those  for  compression  are  increased.  The  action  of  the  combina- 
tion of  the  steel  and  concrete  then  approaches  more  and  more  nearly 
to  that  of  the  flat  arch  with  the  tie  rod  in  the  bottom  in  which  the 
lines  of  compression  arch  more  and  more  from  the  end  upward  to- 
ward the  center,  for  those  bars  at  least  which  are  arranged  in  the 
bottom  of  the  beam  thruout.  If,  however,  there  are  a  number  of 
bars  in  the  beam,  part  of  which  are  in  the  bottom  layer  thruout 
and  other  bars  which  bend  upward  from  the  quarter  point  or  toward 
the  end  of  the  beam,  then  we  have  an  arrangement  such  that  the 
bond  shears  from  the  bent  up  bars  toward  the  end  of  the  beam  bring 
the  upper  layers  of  the  concrete  toward  the  end  into  direct  compres- 
sion and  co-action,  which  the  bars  in  the  bottom  of  the  beam  cannot 
effectively  do.  Hence  in  the  beams  in  which  there  are  bent-up 
bars  a  higher  percentage  of  steel  can  be  used  without  overstraining 
the  beam  by  indirect  tension  than  is  the  case  where  the  metal  is 
placed  in  the  bottom  layer  thruout. 

Stirrups  in  the  form  of  small  rods  extending  vertically  from  the 
bottom  of  the  beam  to  the  top  and  crossing  the  lines  of  indirect  stress 
at  an  angle  assist  in  resisting  indirect  stresses  and  add  materially  to 
the  shearing  strength  of  the  beam.  This  additional  strength  afforded 
by  the  stirrups  has  been  very  thoroly  investigated  by  the  German 
firm  of  Wayss  and  Freytag,  and  the  results  have  been  given  quite 
completely  in  "Der  Eisenbetonbau;'  by  Emil  Morsch,  which  has 
been  translated  by  E.  P.  Goodrich,  and  published  by  the  Engineering 
News  Publishing  Co.  under  the  title  "Concrete  Steel  Construction" 


118 


Fig.  43.     Cottancin  Beam  Construction  in  the  Church  of  St.  Jean  de  Montmartre 


COTTANyiN    CONSTRUCTION  119 

in  which  the  reader  will  find  an  exhaustive  discussion  of  these  valu- 
able tests. 

The  dissipation  of  energy  in  the  combination  of  concrete  and 
steel  in  the  beam  and  the  inequality  of  the  deflections  after  the  applied 
loads  stress  the  steel  beyond  4000  Ibs.  have  been  already  noted  as 
due  to  the  overstrain  or  checking  of  the  concrete  largely  by  indirect 
tensions  or  bond  shears.  An  arrangement  of  the  metal  in  the  web 
may  be  made  which  so  reinforces  it  that  there  is  no  loss  of  energy 
from  this  cause  and  in  such  manner  that  the  bending  resistance  of 
the  steel  may  be  largely  increased  if  not  indeed  doubled.  Since 
every  pound  of  pull  on  the  bar  is  induced  by  bond  shear  it  follows 
that  the  energy  stored  by  internal  work  of  indirect  stress  in  the 
concrete  is  equal  to  that  stored  in  the  steel.  When,  however,  yield- 
ing of  the  concrete  occurs  and  energy  is  dissipated  thereby  the  stress 
in  the  steel  is  found  by  experiment  to  largely  increase  until  the 
stress  in  the  steel  is  substantially  that  which  we  compute  on  the 
basis  that  the  entire  tensile  resistance  is  furnished  by  the  steel 
element. 

In  the  Cottangin  construction  the  entire  web  of  the  beam  (which 
is  very  thin)  is  reinforced  by  a  net  work  of  small  rods  in  such  manner 
that  this  yielding  of  the  concrete  is  prevented  by  thoro  dissemination 
of  steel  both  vertically  and  horizontally,  and  any  deformation  of  the 
concrete  brings  this  net-work  of  steel  into  action  in  such  manner 
that  the  indirect  stresses  from  the  bond  shears  of  the  vertical  and 
horizontal  rods  coact  with  each  other,  and  deflection  of  the  beam  is 
for  all  loads  light  and  heavy  more  nearly  proportional  to  the  loads. 

Marsh  in  his  work  on  Reinforced  Concrete  has  called  particular 
attention  to  this  deportment  of  the  Cottangin  beam  construction 
which  he  considers  is  based  upon  the  theory  of  prevention  of  molecu- 
lar deformation  by  supplying  resistances  of  the  reverse  kind  to 
stresses  on  small  particles  which  produce  notably  good  results  and 
very  light  structures. 

The  mechanical  operation  of  the  Cottangin  beam  will  become 
apparent  from  more  complete  consideration  of  the  action  of  the 
bond  stresses  and  the  storage  of  energy  of  the  internal  work  in  the 
web  of  the  beam  by  the  indirect  stresses  resulting  from  bond  shear 
in  a  scientific  and  stable  manner  which  it  is  impossible  to  effect  in 
a  one  way  reinforced  construction  lacking  thoro  dissemination  of 
the  steel  to  resist  web  stress  in  such  manner  that  the  low  tensile 
resistance  of  the  concrete  may  not  be  over-taxed. 


120 


COTTAN(,'IN    CONSTRUCTION 


The  carrying  capacity  of  the  Cottangin  beam  as  Marsh  has  pointed 
out  gives  results  not  accounted  for  on  the  basis  of  the  usual  theory 
but  which  may  be  accounted  for  readily  when  we  consider  the  scien- 
tific manner  of  reinforcing  the  web  so  that  the  indirect  stresses  are 
provided  for  in  a  manner  which  does  not  over-tax  the  concrete  and 
so  the  loss  of  energy  accompanying  the  inelastic  deformation  of  the 
ordinary  beam  is  avoided. 

The  same  general  principle  differently  applied  is  brought  into 


Fig.  44.     Cottancin  Beam  Construction  in  the  Church  of  St.  Jean  de  Montmarte. 


play  in  multiple  way  slabs,  where  the  arrangement  of  the  metal 
is  such  that  the  energy  stored  by  indirect  stress  is  stored  in  a  depend- 
able or  stable  manner. 

In  column  construction  this  principle  may  be  employed  by  a 
proper  combination  of  hooping  and  vertical  steel.  If  hooping  only 
is  used,  the  fireproofing  or  outside  shell  scales  early  under  test, 
whereas  if  the  combination  of  vertical  steel  and  hooping  in  proper 
proportion  and  arrangement  is  used  85  percent  of  the  ultimate 
strength  of  the  specimen  may  be  developed  before  scaling  and  chipping 
of  the  outside  shell  occurs.  It  is  to  be  borne  in  mind  that  the  checking 


BOND    SHEAR 


121 


or  cracking  under  load  is  an  inelastic  deformation  accompanied  by 
the  dissipation  of  energy  and  followed  by  the  development  of  new 
energy  by  the  downward  motion  of  the  load  thru  increased  deforma- 
tion before  equilibrium  can  be  reestablished.  Thus  the  hooped 
column,  without  verticals,  is  objectionable  on  the  ground  of  excessive 
deformation,  whereas  with  a  proper  combination  of  vertical  steel 
and  hooping  this  excessive  deformation  is  prevented  since  the  energy 
of  internal  work  is  stored  in  a  dependable  manner  thru  a  scientific 
arrangement  of  the  reinforcement  which  induces  coaction  of  the 
indirect  stresses  in  the  concrete  matrix  which  are  generated  on  the 
one  hand  by  the  bond  shear  between  the  matrix  and  the  verticals 


£ 

B 

^    ^ 

X 

X 

X 

X 

2 

X 

X 

y 

X 

2 

\ 

1 

g 

J  1 

% 

v    Is 

Fig.  45. 


and  on  the  other  between  the  hooping  and  the  matrix,  as  will  be  ex- 
plained more  at  length  in  succeeding  pages. 

A  fundamental  principle  of  concrete  design  from  the  standpoint 
of  work,  may  be  stated  in  the  following  words: 

The  most  perfect  design  is  one  in  which  the  potential  energy  of 
internal  work  is  stored  in  the  most  stable  manner.  Such  a  design, 
from  the  commercial  standpoint,  will  be  economical  in  the  quantity 
of  material  required  and  for  a  given  depth  of  beam  or  slab  will  carry 
a  greater  load  with  less  deflection,  and  may  be  expected  to  prove 
most  satisfactory  under  severe  conditions  of  repeated  loading. 

Indirect  Stresses  Generated  by  Bond  Shear  and.  the  Laws  of  Distribution 

Governing  These  Mechanical  Actions 

19.  Bond  Shear  in  Blocks.  Let  R  Fig.  45  represent  a  cylindrical 
metallic  rod  embedded  in  a  block  of  concrete  B  B  in  which  it  has 


122  BOND  SHEAR  IN  BLOCKS 

been  cast.  Let  it  rest  on  a  ring  shaped  support  SS  having  a  central 
apperture  somewhat  larger  than  the  rod.  Then  the  weight  II' 
applied  to  the  rod  will  induce  a  bond  shear  at  the  surface  of  the  rod, 
whose  intensity  will  depend  upon  several  factors,  among  which  will 
be  the  magnitude  of  the  weight  W,  the  diameter  D  of  the  rod,  the 
length  L  of  the  embedment,  and  the  distance  of  the  point  considered 
from  the  beginning  of  the  embedment.  Other  things  being  equal, 
the  intensity  of  the  bond  shear  will  be  greatest  at  the  point  of  em- 
bedment nearest  the  point  of  application  of  W,  whether  the  rod 
be  in  tension  or  in  compression.  Since  the  concrete  which  surrounds 
the  rod  is  in  a  state  of  shearing  stress  along  the  surface  of  the  rod, 
it  is  in  a  state  of  stress  that  may  also  be  denned  otherwise  by  saying 
that  there  is  indirect  or  induced  tension  in  the  concrete  along  lines 
sloping  at  45  towards  the  axis  of  the  rod,  represented  on  Fig.  45  by 
the  arrows  with  heads  at  the  surface  of  the  rod,  and  also  compression 
induced  in  the  concrete  along  lines  sloping  at  45°  away  from  the 
axis,  the  intensity  of  each  of  these  induced  stresses  at  the  surface 
of  the  rod  being  the  same  as  that  of  the  bond  shear  at  the  surface 
and  growing  less  at  greater  distances  from  the  axis  nearly  inversely 
as  the  distance.  The  arrows  may  be  regarded  as  representing  lines 
of  force  or  stress  in  the  concrete  in  such  a  manner  that  the  intensity 
of  the  stress  is  proportional  to  their  nearness  to  each  other.  The  total 
bond  shear  between  rod  and  concrete  amounts  to  W,  and  is  so  dis- 
tributed on  the  surface  of  the  rod  that  it  is  small  at  the  point  of  em- 
bedment most  distant  from  W  and  is  greatest  at  the  points  nearest 
to  W.  The  bond  shear  at  any  point  is  the  increment  of  the  tension 
in  the  rod  at  that  point  and  there  is  no  element  of  tension  in  the  rod 
which  is  not  balanced  and  transmitted  to  the  concrete  by  a  corres- 
ponding equal  element  of  bond  shear. 

The  block  tends  to  decrease  in  length  by  reason  of  the  load  thus 
transmitted  to  it  at  the  same  time  that  the  part  of  the  rod  embedded 
in  the  concrete  tends  to  increase  in  length,  so  that  the  point  where 
the  first  slipping  will  occur  is  at  the  point  nearest  to  W.  But  if 
no  actual  slipping  occurs  the  distortion  is  greatest  at  this  point. 
This  is  the  reason  that  the  greatest  intensity  of  bond  shear  occurs  at 
this  point. 

In  the  space  where  the  arrows  are  represented  it  might  at  first 
be  thought  that  the  radial  components  of  the  induced  tensions  and 
compressions  would  neutralize  each  other.  But  such  is  not  the 
case  because  it  is  a  fundamental  property  of  stress  in  any  material 
that  a  shear  in  any  plane  (as  for  example  the  vertical  plane  in  this 


BOND    SHEAR    IN    A    SPLICE 


123 


case)  in  order  to  maintain  equilibrium  is  necessarily  accompanied 
by  an  equal  opposite  shear  on  a  plane  at  right  angles  with  it,  so  that 
these  induced  vertical  stresses  act  to  produce  radial  shears  of  equal 
intensity  on  each  horizontal  plane. 

This  horizontal  shear  in  the  block  diminishes  somewhat  as  the 
radius  increases  but  does  not  vanish  until  we  reach  the  inner  edge 
of  the  ring  SS.  Beyond  the  inner  edge  of  SS  a  vertical  compression 
exists  in  the  block  which  causes  a  pressure  upon  the  ring  SS,  whose 
intensity  is  greatest  at  the  inner  edge  of  SS,  but  which  diminishes 
beyond  the  inner  edge  of  SS  in  a  ratio  which  need  not  here  be  con- 
sidered. 

The  matter  of  importance  in  this  discussion  is  the  manner  in 
which  the  stress  is  transmitted  thru  the  concrete  around  the  rod, 


Fig.  46 

where  the  concrete  has  lines  of  stress  in  it  similar  to  those  in  the  web 
of  a  built  beam  which  cause  the  two  flanges  to  coact  as  do  the  rod 
and  exterior  part  of  the  block  in  this  case. 

The  induced  tensions  which  have  just  been  described  are  the  same 
as  those  frequently  designated  as  diagonal  tensions.  Failure  arising 
from  excessive  stress  of  this  kind  would  evidently  give  rise  to  cracks 
nearly  at  right  angles  to  the  arrows  representing  the  induced  tensions, 
i.  e..to  cracks  along  the  lines  of  compression,  but  such  cracks  could 
not  occur  without  excessive  elongation  along  the  rod,  nor  would 
they  be  expected  to  occur  in  a  block  at  all  under  ordinary  circum- 
stances, because  they  would  be  preceded  by  slipping  of  the  bond, 
after  which  the  diagonal  shear  would  be  relieved  where  the  slipping 
took  place  and  transferred  to  some  point  further  along  the  rod. 

20.  Bond  Shear  in  Splices.  Let  Fig.  46  represent  a  splice  in  a 
belt  of  reinforcing  rods  with  laps  of  length  L.  The  entire  effective- 
ness of  the  splice  depends  upon  the  bond  shear  at  the  surfaces  of  the 


124  INDIRECT    STRESSES    IN    BEAMS 

belt  rods  in  which  the  action  of  the  embedment  is  in  many  particu- 
lars like  that  already  discussed,  but  has  several  new  features  especially 
by  reason  of  the  dissymmetry  of  the  embedment  laterally,  altho 
longitudinally  it  has  perfect  symmetry  such  as  the  block  did  not 
possess. 

The  lateral  arrangement  is  such  that  the  stresses  in  the  concrete 
between  the  rods  on  the  sides  where  they  are  nearest  together  is  of 
much  greater  intensity  than  elsewhere,  so  that  the  diagram  of  the 
stresses  about  a  rod  would  be  much  like  that  shown  in  Fig.  45,  except 
that  on  one  side  of  the  rod  the  arrows  would  cover  the  space  very 
closely,  while  on  the  other  side  the  arrows  would  be  few  and  far 
between.  Moreover  the  successive  arrows  would  be  just  as  near  each 
other  at  one  end  of  L  as  at  the  other  end,  but  nearer  together  at  each 
end  than  nearer  the  middle  of  L.  The  bond  shear  would  fail  first 


w\ 


Fig.  47. 

on  each  rod  at  that  end  of  L  furthest  from  the  end  of  the  rod.  The 
lines  of  force  in  the  horizontal  plane  of  the  belt  would  be  as  just 
described,  but  lines  of  force  which  start  out  from  the  rods  in  a  direc- 
tion above  or  below  that  plane  would  curve  around  spirally  from  one 
rod  to  the  other,  for  each  line  of  force  must  have  one  end  on  one  rod 
and  the  other  on  some  other  rod  near  by. 

The  strength  of  the  splice  depends  upon  the  bond  shear  at  the 
surfaces  of  the  rods,  and  on  resistance  to  the  induced  tensions  and 
compressions  along  the  lines  of  stress  aff.orded  by  the  concrete. 
The  integrity  of  the  concrete  embedment  under  the  action  of  these 
stresses  is  essential  to  the  transmission  of  force  along  the  splice,  and 
were  the  limiting  values  of  either  bond  shear,  indirect  tension,  or 
compression,  exceeded  in  the  concrete  the  splice  would  fail. 

21.  Bond  Shear  in  Beams.  Let  Fig.  47  represent  a  reinforced 
concrete  beam  of  length  L  supported  at  the  ends  and  loaded  with 
two  equal  weights  W  each  placed  symmetrically  at  the  same  dis- 
tance, nL  from  the  end  nearest  to  it.  Then  the  total  vertical  shear 
at  any  point  between  the  equal  weights  is  zero  and  the  bending 
moment  is  nWL,  while  at  any  point  between  either  load  W  and  the 


BOND  SHEAR  IN  BEAMS  125 

end  nearest  to  it  the  total  vertical  shear  is  W  and  the  moment  is 
Wx,  where  x  is  the  distance  of  the  point  from  the  end  nearest  to  it. 
In  such  a  beam  where  there  is  no  vertical  shear  at  any  point  of  the 
middle  segment  there  is  no  horizontal  shearing  stress  at  any  point 
of  this  segment,  consequently  the  bond  shear  stress  is  zero  thruout 
this  segment. 

Let  N  represent  the  position  of  the  neutral  axis.  The  concrete 
above  N  is  in  direct  compression  and  that  below  N  is  subject  to 
direct  tension  along  horizontal  lines  none  of  which  are  repre- 
sented in  Fig.  47.  The  entire  length  of  the  reinforceing  rod  R  is 
under  tension,  the  central  segment  between  the  applied  weights 
being  under  a  uniform  tension  thruout  which  will  be  determined 
ultimately  by  the  bending  moment  WL,  for  if  the  weights  W  are 
sufficiently  great  or  other  contingencies  such  as  rapid  setting,  or  long- 
continued  loading  have  occurred,  enough  vertical  cracks  will  have 
developed  at  the  bottom  of  the  beam  in  this  segment  to  have  prac- 
tically destroyed  its  direct  tensile  resistance  to  horizontal  force. 
Provided  there  is  sufficient  horizontal  reinforcement  to  safely  resist 
this  horizontal  tension  and  the  concrete  above  N  is  sufficient  to  resist 
the  compression  the  existence  of  vertical  cracks  in  this  segment  of 
the  beam  at  the  bottom  is  not  to  be  regarded  as  a  sign  of  structural 
weakness  or  ultimate  failure  of  the  beam.  These  may  under  these 
conditions  be  regarded  as  harmless  characteristic  cracks.  But  in 
case  the  reinforcement  is  insufficient,  yielding  will  occur  in  this  part 
of  the  beam  where  the  moment  is  greatest  and  such  yielding  will 
cause  failure. 

In  the  end  segments  of  the  beam  the  moment  of  the  bond  shears 
S  per  unit  of  length  of  reinforcement  represented  by  Sjd  must  be  equal 
to  the  increment  of  the  bending  moment,  i.  e.  equal  to  the  total 
vertical  shear  W  provided  it  be  assumed  that  the  reinforcement 
furnishes  the  entire  direct  tensile  resistance  and  the  concrete  none. 
Assuming  that  case  for  safe  design,  the  bond  shear  is  represented 
in  Fig.  47,  in  the  same  manner  as  in  Fig.  45,  viz :  the  arrows  with  heads 
at  the  reinforcing  rods  represent  the  lines  of  indirect  tension  in  the 
concrete,  and  those  with  heads  at  the  neutral  axis  N  represent  the 
lines  of  compression  induced  by  the  bond  shear.  The  concrete  is 
abundantly  able  to  afford  the  necessary  resistance  to  the  indirect 
compression,  but  may  be  entirely  unable  to  afford  the  necessary 
resistance  to  the  indirect  or  diagonal  tension,  which  may  conse- 
quently cause  cracks  perpendicular  to  its  direction,  that  is  along  the 
lines  of  indirect  compression,  and  at  the  same  time  produce  rupture 


126  DISTRIBUTION    OF    INDIRECT    STRESS 

of  the  bond  shear.  In  beams  so  heavily  reinforced  as  to  prevent 
central  yielding,  failure  due  to  rupture  of  diagonal  tension  and  rup- 
ture of  bond  along  the  steel  ordinarily  occurs  under  overload.  These 
two  accompany  each  other  because  they  are  of  practically  equal 
intensity  as  noticed  before,  and  the  cracks  due  to  slipping  of  the 
steel  are  continuous  with  those  due  to  diagonal  tension.  Additional 
reinforcement  to  resist  diagonal  tension  consists  of  stirrups  and  the 
like,  which  are  enabled  to  do  this  by  their  own  secondary  system  of 
bond  shears  and  anchorage. 

The  system  of  induced  or  diagonal  tensions  which  have  just  been 
treated  are  necessarily  modified  somewhat  in  amount  and  direction 
by  their  combination  with  the  direct  horizontal  tensions  in  the  con- 
crete due  to  the  bending  where  these  direct  tensions  are  not  elimi- 
nated by  vertical  cracks.  But  the  existence  and  effectiveness  of 
the  induced  tensions  depending  on  the  bond  shear  remains  practically 
unimpaired  so  long  as  the  bond  is  intact. 

It  should  be  noticed  that  the  distribution  of  the  bond  shear  on 
the  surface  of  the  reinforcement  must  be  more  intense  on  the  upper 
sides  of  the  rods  by  reason  of  the  greater  rigidity  of  the  concrete 
on  that  side,  due  to  its  backing  above,  and  lack  of  backing  below. 

In  case  of  a  beam  carrying  a  load  distributed  differently  from 
that  assumed  in  Fig.  47,  there  may  be  no  segment  of  any  finite  length 
where  the  shear  vanishes,  but  such  a  segment  of  infinitesimal  length 
will  exist  wherever  the  bending  moment  is  constant,  i.  e.  wherever 
it  is  a  maximum  or  a  minimum.  Moreover  the  total  values  of  the 
bond  shear  per  unit  of  length  of  the  reinforcement,  multiplied  by 
jd  will  equal  the  total  vertical  shear  in  case  the  reinforcement  takes 
the  entire  direct  tension.  In  such  a  case  the  characteristics  of  the 
middle  and  end  segments  of  Fig.  47,  merge  into  each  other  somewhat, 
in  a  manner  not  difficult  to  understand. 

It  will  be  readily  seen  that  the  manner  in  which  the  stresses  are 
distributed  in  any  case  of  reinforcement  depends  entirely  upon  the 
applied  forces  and  the  distribution  of  the  rigidities  due  to  the  amount 
and  distribution  of  the  steel,  but  that  the  coaction  of  concrete  and 
steel  is  brought  about  by  stresses  communicated  from  one  to  the 
other  thru  the  medium  of  bond  shear  which  is  vital  to  the  discussion 
of  any  such  question,  and  that  together  the  steel  and  concrete  form  a 
combination  or  machine  which  has  properties  as  a  whole  which  are 
different  from  the  properties  of  either  of  the  constituents.  What 
those  properties  may  be  in  any  case  must  be  determined  from  a 
careful  analysis  of  the  particular  case  under  consideration. 


BOND    SHEAR,    IN    SLABS 


127 


It  should  be  further  noticed  that  the  lines  of  tension  that  origi- 
nate or  are  generated  in  the  bond  shear  at  any  given  element  of  the 
surface  of  a  rod  are  to  be  thought  of  as  physically  independent  of 
the  lines  of  compression  originating  at  the  same  point  in  such  fashion 
that  the  tension  starting  at  a  given  point  a  say,  is  held  in  equilibrium 
at  the  other  end  of  that  line  by  a  tension  at  some  point  6,  while  the 
compression  which  originates  at  the  same  point  a,  is  resisted  and 
held  in  equilibrium  by  a  resistance  to  compression  at  some  entirely 
different  point  c.  Moreover,  it  may  be  that  b  and  c  are  on  the 
surfaces  of  entirely  different  bodies,  as  frequently  occurs  in  case  of 
splices  in  a  belt. 


22.  Bond  Shear  in  Slabs.  Let  Fig.  48  represent  the  crossing  of 
two  reinforcing  rods  under  tension  in  a  multiple  way  reinforcement 
as  for  example  in  a  slab  where  the  shearing  stress  of  the  bond  resists 
any  tendency  of  the  rods  to  slip  longitudinally  in  the  direction  of 
the  long  arrows.  Then  the  mutual  action  of  the  lines  of  tension  and 
compression  arising  from  the  bond  shear  is  that  represented  in  the 
figure  where  the  arrows  with  points  against  the  rods  represent  lines 
of  tension  and  those  pointed  away  from  the  rods  lines  of  compression. 
There  exist  other  lines  due  to  the  bond  shears  besides  those  here 
represented  which  are  similar  in  their  disposition  and  action  to  those 
found  in  beam  action  which  has  been  already  discussed.  But  such 
other  lines  are  here  omitted  from  discussion  because  by  the  principle 
of  rigidities  the  lines  which  are  here  represented  as  affording  short 
and  direct  connections  thru  the  concrete  are  necessarily  the  ones 
which  include  the  predominant  action  of  the  bond  shear.  This 
predominant  action  is  here  seen  to  resemble  in  its  general  nature  that 


128  PRINCIPLE    OF    RIGIDITIES 

of  a  splice,  but  has  laws  of  distribution  peculiarly  its  own  by  reason 
of  the  relative  situation  of  the  rods  in  the  concrete,  which  is  such 
that  the  intensity  of  the  stress  developed  is  greatly  intensified  in 
the  immediate  vicinity  of  the  point  of  contact  of  the  rods. 

The  principle  of  rigidities  here  referred  to  as  determining  the 
predominant  action  is  one  susceptible  of  exceedingly  simple  illus- 
tration yet  one  whose  final  implications  and  applications  are  of  such 
a  nature  as  to  make  the  layman  think  for  the  instant  that  it  is  im- 
possible to  establish  with  certainty  the  definite  conclusions  that  can 
nevertheless  be  at  once  conclusively  demonstrated. 

To  illustrate  this  point,  let  there  be  two  springs  or  spring-balances, 
one  of  which  is  twice  as  stiff  or  rigid  as  the  other,  and  if  side  by  side 
they  together  support  a  given  weight,  the  more  rigid  spring  will 
carry  the  larger  proportion  of  the  load,  two  pounds  to  each  pound 
of  the  other.  Just  so  in  any  structure  where  there  are  several  parts 
of  the  structure,  each  of  which  is  to  carry  its  part  of  the  load  the 
question  as  to  how  the  load  will  be  divided  between  them  is  deter- 
mined by  their  relative  rigidity  or  stiffness,  that  part  which  lacks 
stiffness  and  yields  easily  will  carry  little  of  the  load  and  will  leave 
the  brunt  of  the  work  to  those  parts  that  are  rigid  and  unyielding. 
Furthermore,  since  the  deformation  multiplied  by  the  mean  force 
(or  half  the  final  load)  is  the  work  done  during  deformation,  it  is 
evident  that  for  a  given  applied  load  the  more  rigid  parts  carry  their 
part  of  the  load  with  less  deformations  than  the  other  parts  could 
carry  it.  Hence  the  work  of  deformation  is  less  by  this  arrangement 
than  it  would  be  by  supposing  less  rigid  parts  to  carry  it.  This  is 
the  principle  of  least  work,  which  states  that  of  all  the  various  ways 
in  which  a  given  elastic  structure  could  be  supposed  to  be  deformed 
in  carrying  a  load,  the  way  in  which  it  will  actually  be  deformed  is 
that  in  which  the  load  will  perform  the  least  work  in  deforming  the 
structure. 

Fig.  48,  showing,  as  it  does,  merely  a  ground  plan  of  the  lines  of 
force  cannot  represent  fully  the  distribution  of  the  points  of  attach- 
ment of  the  lines  of  force  which  are  generated  all  over  the  surfaces 
of  the  rods  and  start  out  everywhere  like  the  quills  of  a  hedgehog, 
in  greatest  number  at  points  where  the  two  rods  are  nearest  to  each 
other,  until  in  the'  immediate  neighborhood  of  the  point  of  contact 
between  the  rods,  the  concrete  grips  the  rods  and  holds  them  with  a 
firmness  and  strength  that  would  seem  incredible  did  not  experience 
demonstrate  its  safety  and  reliability. 

It  is  this  dominant  action  which  is  greatly  increased  in  four  way 


INTERACTION    BETWEEN    RODS  129 

reinforcement  which  distinguishes  slab  action  from  beam  action, 
and  brings  into  play  the  dependable  indirect  stresses  in  the  concrete 
in  contradistinction  from  the  direct  tensile  stresses  due  to  beam 
action.  A  failure  to  understand  the  physics  and  mechanics  of  this 
predominant  action  of  slab  reinforcement  as  depending  on  the 
fundamental  role  which  bond  shear  plays  in  this  case  has  prevented 
those  who  have  discussed  this  subject  from  grasping  the  real  relations 
that  here  exist.  It  has  been  denied  most  vehemently  that  it  is 
possible  for  the  stress  in  any  rod  to  influence  in  any  way  the  stress 
in  any  rod  that  crosses  it,  whereas  such  action  is  not  only  unavoid- 
able when  bond  shear  exists  on  both,  but  is  the  key  to  the  situation 
and  explains  the  otherwise  inexplicable  phenomena  here  observed. 
The  mechanism  of  lateral  interaction  between  the  rods  in  a  slab  is 
such  as  to  make  the  slab  have  properties  which  imitate  in  many 
respects  those  of  a  continuous  plate,  and  the  efficiency  of  this  action 
can  be  treated  as  in  such  a  plate  by  help  of  a  coefficient. 

Why  should  anyone  deny  that  a  reinforced  slab  acts  in  a  manner 
similar  to  a  continuous  plate?  The  only  basis  for  such  a  denial  is 
the  assertion  that  in  order  to  so  act  the  direct  tensile  strength  of  the 
concrete  will  have  to  be  relied  on  which  cannot  be  relied  on  in  a  beam. 
That  assertion  involves  a  fallacy.  All  that  need  be  relied  on  to  make 
this  action  effective  is  the  indirect  tensile  strength  of  the  concrete 
which  is  certainly  reliable  for  this  purpose  to  the  required  extent 
as  appears  from  the  fact  that  the  bond  does  not  in  fact  suffer  rupture 
in  well  cured  slabs. 

Any  analytical  method  of  dealing  with  the  phenomena  of  plate 
action  or  slab  action  by  which  the  stresses  in  various  directions 
affect  each  other  depends  upon  the  introduction  of  the  so-called 
Poisson's  ratio  to  take  account  of  this  interaction,  so  that  the  num- 
erous and  emphatic  denials  in  technical  papers  of  the  applicability 
of  this  ratio  to  slab  action  show  in  fact  an  entire  ignorance  or  mis- 
understanding of  what  that  action  is. 

It  may  in  general  be  stated  that  stresses  in  steel  reinforcement, 
especially  in  slab  steel  are  imparted  to  the  steel  by  the  bond  shear 
and  not  by  any  other  kind  of  anchorage  in  the  concrete,  such  as 
occurs  near  the  ends  of  simple  beams.  From  this  it  follows  that  the 
total  stresses  in  the  steel  at  any  point  must  have  existed  previously 
in  the  concrete.  Since  a  given  distributed  force  or  stress  considered 
as  a  single  thing  has  a  certain  magnitude  thruout  its  length,  at  such 
places  as  it  is  present  in  the  concrete  it  is  not  present  in  the  steel 


130  MECHANICS    OF    EMBEDMENT 

and  vice  versa.     The  mean  stress  to  be  carried  by  the  steel  is  con- 
sequently diminished  by  that  in  the  concrete. 

It  comes  about,  therefore,  that  the  stresses  in  the  steel  are  re- 
duced by  the  action  of  the  concrete,  or  the  steel  has  less  stress  to  carry 
by  reason  of  its  embedment  than  if  it  supplied  the  total  tensile 
resistance  itself.  It  is  thus  protected  by  the  action  of  the  embed- 
ment from  some  portion  of  the  tensile  stress  without  calling  upon 
the  embedment  for  such  direct  tensile  resistance  as  is  shown  to  be 
liable  to  failure  in  beams,  but  is  calling  instead  merely  for  such 
indirect  tensile  stresses  as  are  known  to  be  safe  and  dependable 
from  what  we  know  of  splices  and  slabs.  These  two  things  then  are 
the  basis  of  slab  theory:  the  reliability  of  bond  shear  in  slabs  and 
the  reduction  that  it  produces  in  the  stresses  that  without  it  would 
exist  in  the  rods  constituting  the  multiple  way  reinforcement.  As 
seen  above  this  reduction  may  amount  in  certain  cases  to  50  percent 
of  the  total  force  acting.  A  very  instructive  experiment  in  this 
connection  was  made  at  the  Minneapolis  Court  House  of  a  slab  6 
inches  thick,  and  7  by  10  feet  horizontally,  reinforced  in  two  direc- 
tions lengthwise  and  crosswise  with  f  inch  round  rods  8  inches  between 
centers.  When  tested  as  a  beam  supported  at  the  ends,  the  cross- 
wise steel  not  being  subject  to  bond  shear  or  tensile  stress,  did  not 
increase  the  resistance  to  flexure,  nor  prevent  the  longitudinal 
steel  from  receiving  the  full  effect  of  the  bending  moments. 

23.  Mechanics  of  Embedment.  The  mechanics  of  reinforced 
concrete  under  flexure  may  be  summarized  as  follows: 

The  co-operation  or  combined  action  of  the  two  materials,  con- 
crete and  steel,  to  resist  bending,  depends  solely  on  the  bond  between 
the  two,  which  has  been  discussed  briefly  in  a  preceding  article. 

In  the  case  of  plain  rods,  this  bond  is  in  reality  a  shrinkage  grip 
which  prevents  the  steel  from  sliding  thru  the  hardened  matrix 
in  which  it  is  embedded,  and  the  resistance  afforded  by  this  bond 
is  subject  to  well  defined  laws  which  may  be  stated  as  follows: 

The  bond  shear  is  zero  wherever  the  tension  in  the  steel  is  con- 
stant. It  passes  thru  zero  where  the  increment  of  the  moment 
passes  thru  a  maximum  or  minimum.  It  must  be  depended  upon 
whether  the  reinforcement  is  in  one  direction  only  as  in  a  beam,  or 
in  multiple  directions  in  the  slab. 

Bond  shear  generates  stresses  emanating  from  the  surface  of  the 
bars  which  may  be  treated  as  lines  of  force.  These  lines  of  force 
follow  the  general  laws  of  distribution  of  force  thru  any  medium, 


INDIRECT    STRESS    IN    THE    SIMPLE    BEAM  131 

that  is,  their  intensity  is  inversely  as  the  square  of  the  distance 
from  the  surface  of  the  steel  on  which  they  are  generated. 

These  general  laws  enable  us  to  investigate  or  follow  out  the  part 
played  by  bond  shear  in  the  mechanics  of  a  slab  or  beam.  In  the 
case  of  a  simple  beam  in  accordance  with  the  law  stated,  the  intensity 
of  the  bond  shear  is  zero  at  the  center  for  uniform  load  and  a  max- 
imum toward  the  end  of  the  beam,  and  it  is  to  the  bond  shear  or  the 
lines  of  force  generated  thereby  to  which  we  may  attribute  the 
difference  in  the  failure  of  an  over  and  an  under  reinforced  beam. 

In  the  case  of  the  beam  with  light  reinforcement,  failure  takes 
place  at  the  center  by  the  yielding  of  the  steel.  With  heavier 
reinforcement,  on  the  contrary,  failure  is  more  liable  to  occur  toward 
the  end  by  indirect  tension  induced  by  the  bond  shear  which  is 
greatest  toward  the  end  of  the  beam  and  which  may  be  resisted  only 
by  the  direct  tensile  strength  of  the  concrete  itself. 

The  deportment  of  the  simple  beam  as  affected  by  the  stresses 
set  up  by  the  bond  shear  is  of  interest.  In  a  newly  cast  beam,  in 
the  preliminary  stages  of  the  loading,  the  stress  in  the  steel  as  deter- 
mined by  the  extensometer  is  much  less  than  that  figured  on  the 
assumption  that  the  steel  only  resists  tension.  In  fact,  it  is  only 
about  half  as  great  as  we  should  compute  the  stress  to  be  on  the  above 
basis,  until  the  steel  is  stressed  up  to  four  or  five  thousand  pounds 
per  square  inch.  When  this  point  has  been  reached,  there  is  a  rapid 
increase  in  the  stress  in  the  steel  with  no  corresponding  increase 
in  the  load  until  when  the  steel  is  stressed  up  to  twelve  or  fifteen 
thousand  pounds  per  square  inch,  the  concrete  has  relieved  itself 
of  a  large  portion  of  its  tensile  resistance  and  the  measured  stress 
in  the  steel  corresponds  closely  to  the  computed  stress  in  the  steel, 
assuming  the  steel  not  to  be  assisted  by  the  concrete  in  tension. 

With  the  slab  reinforced  in  two  directions,  however,  the  phe- 
nomenon differs  from  that  observed  in  the  beam.  Take  for  example 
the  case  of  the  slab  reinforced  in  two  directions,  bent  in  such  a  man- 
ner that  the  rods  in  both  directions  are  brought  into  tension  at  the 
same  time.  The  indirect  stresses  generated  by  the  two  sets  of  rods 
will  under  this  condition  react  upon  each  other.  Since  the  lines 
of  force  diverge  from  each  rod  they  may  meet  and  coact  thru  the 
concrete  as  a  medium  of  transmission  of  the  stress,  which  is  not 
possible  in  the  beam  with  one  way  reinforcement,  since  in  the  beam 
these  stresses  cannot  coact  with  each  other,  there  being  one  kind 
only  and  not  two  kinds  acting  in  different  directions.  This  funda- 
mental difference  in  the  stress  induced  by  the  bond  shear  in  the  case 


132  LAWS    OF    INDIRECT    STRESS 

of  a  beam  from  that  in  a  slab  renders  the  two  types  of  structure 
mechanically  different  and  necessitates  their  treatment  in  a  manner 
which  takes  into  consideration  the  difference  in  the  mechanical 
operation  of  the  indirect  stresses  referred  to. 

A  crack  would  not  materially  interfere  with  the  operation  of 
these  indirect  stresses  in  a  multiple  way  reinforced  slab,  because 
the  paths  of  the  lines  of  force  will  still  be  able  to  find  other  passage 
ways  such  as  avoid  the  cracks.  But  a  crack  in  a  beam  which  is 
normal  to  the  direction  of  the  steel  would  intercept  the  indirect 
tensions  induced  by  the  bond  shear  at  the  section  checked,  and 
prevent  the  accumulated  resistances  offered  by  the  indirect  stresses 
from  being  effective  in  direct  resistance  to  moment. 

In  treating  the  combination  of  the  two  materials,  it  has  been 
customary  to  consider  their  combined  action  as  determined  by  the 
elastic  properties  of  each  taken  separately,  namely  by  the  ratio  of 
the  modulus  of  elasticity  of  the  concrete  in  compression  and  tension 
to  the  modulus  of  elasticity  of  the  steel  in  tension  and  compression. 
In  a  homogeneous  elastic  slab  such  as  steel  in  the  form  of  a  plate 
there  must  needs  be  taken  into  consideration  in  addition  to  the 
modulus  of  elasticity  of  the  metal  in  tension  and  compression  in  one 
direction,  the  additional  coefficient  or  modulus  of  lateral  deformation 
known  as  Poisson's  ratio.  This  ratio,  or  lateral  effect  in  a  combina- 
tion of  steel  and  concrete  which  is  sufficiently  fine  grained  to  be 
regarded  as  acting  as  a  homogeneous  material,  as  is  the  case  with 
reinforced  concrete,  cannot  be  correctly  considered  as  an  elastic 
property  of  either  the  concrete  or  the  metal,  but  on  the  contrary 
must  be  treated  as  a  coefficient  expressing  the  efficiency  of  the  lateral 
action  of  the  indirect  stresses  induced  by  the  bond  shear  in  the  case 
of  multiple  way  reinforcement  in  the  slab,  which  coefficient,  for  the 
reasons  above  explained,  must  be  zero  in  the  case  of  the  beam  type 
with  reinforcement  in  but  one  direction  or  in  case  of  the  slab  in  which 
the  reinforcement  under  strain  runs  in  but  one  direction  only. 
Altho  transverse  reinforcement  may  be  introduced  in  a  beam  it  can 
perform  no  useful  function  in  reducing  the  stress  on  the  carrying- 
rods  since  the  indirect  stress  induced  by  one  series  of  rods  under 
stress  cannot  converge  to  react  upon  another  set  of  rods  not  under 
stress,  but  can  react  with  that  set  of  rods  only  when  both  are  gen- 
erating indirect  lines  of  force  from  bond  shear. 

24.  Bond  With  Deformed  Bars.  Morsch  discusses  the  action 
of  deformed  bar  reinforcement  thus:* 


Concrete  Steel  Construction  p.  17. 


BOND    WITH    DEFORMED    BARS  133 

"In  America  various  forms  of  reinforcement  are  employed 
all  of  which  are  designed  to  prevent  slipping  of  the  rod  in  the 
concrete.  In  the  Ransome  rod,  this  is  secured  by  twisting  the 
square  steel  bar;  in  the  Johnson  bar,  elevations  on  the  surface  of 
the  rods  are  produced  in  the  rolling;  and  the  Thacher  or  knotted 
bar  is  provided  with  swellings,  while  maintaining  a  constant  sec- 
tional area.  These  'knots'  may  well  have  the  desired  effect  when 
the  rod  is  anchored  in  a  large  mass  of  concrete,  but  they  will  act 
in  an  opposite  manner  in  the  small  stems  of  T-beams,  especially 
at  their  bottoms,  where  they  will  have  a  splitting  effect  and  thus 
cause  premature  failure  of  bond.  It  will  be  shown  later  that  the 
adhesion  in  the  case  of  ordinary  round  rods  with  hooked  ends  is 
ample  to  transfer  all  actual  stresses,  and  furthermore,  the  arrange- 
ment of  the  principal  reinforcement  may  be  so  designed  with 
respect  to  the  shearing  stresses  that  no  occasion  should  arise  to 
make  up  any  deficiency  through  the  use  of  those  costly  special  bars." 

As  regards  the  mechanics  of  embedment,  the  effect  of  the  "  knots" 
as  Morsch  terms  them,  would  cause  slight  irregularities  in  the  inten- 
sity of  the  indirect  stresses,  assuming  that  there  was  any  over-strain 
tending  to  cause  the  bar  to  slip  and  to  bring  into  action  the  mechan- 
ical bond.  This  variation  from  the  condition  of  uniformity  of 
bond  shear  with  the  plain  bars  would  not  be  material  under  loads 
producing  stresses  not  exceeding  those  which  are  safe,  tho  the  dis- 
advantage suggested  by  Morsch  in  the  case  of  the  thin  ribs  might 
be  looked  for  under  loads  approaching  the  ultimate  strength. 


134 


CHAPTER  IV. 

BEAM  ACTION  AND  SLAB  ACTION  COMPARED  THROUGH    APPLI- 
CATION OF  THE  LAWS  OF   BOND   SHEAR   AND 
THE  THEORY  OF  WORK. 

1.  Introductory.  The  preceding  discussion  of  the  mechanics  of 
the  indirect  stresses  generated  by  bond  shear  indicates  the  laws 
which  govern  the  distribution  of  metal  required  in  order  to  secure 
effective  continuous  p'ate  action  in  type  IV  floors,  i.  e.  flat  slabs 
supported  by  columns. 

At  the  diagonal  center  of  a  panel,  the  moment  of  the  applied  forces 
passes  through  a  maximum  and  the  bond  shear  accordingly  passes 
through  zero,  hence  an  arrangement  of  narrow  strips  of  reinforce- 
ment on  diagonal  lines  from  column  to  column  is  of  no  utility  in 
securing  lateral  efficiency  by  the  operation  of  indirect  stresses.  Wide 
spreading  reinforcement  covering  substantially  the  area  between 
lines  of  inflection  is  the  prime  requisite  for  efficiency. 

The  force  of  this  remark  will  be  better  appreciated  from  a  de- 
tailed consideration  of  the  moment  curve  of  the  circular  suspended 
plate  under  uniform  load. 

The  area  of  a  floor  of  this  type  may  be  subdivided  according  to 
deformation  into  three  divisions : 

a.  Cantilever  area  about  the   columns   convex  upward  and  ap- 
proaching spherical  curvature  for  equal  column  spacing. 

b.  Suspended  circular  plate  concave  upward  about  the  diagonal 
center  of  the  panel. 

c.  Saddle   shaped    areas   between   the    cantilever   areas   and   the 
suspended  central  plate. 

Considering  the  suspended  plate,  the  moment  curve  differs  from 
the  parabolic  curve  of  a  beam  uniformly  loaded  in  that  the  center 
of  gravity  of  that  part  of  the  load  transferred  to  the  support  is  nearer 
to  the  support  than  in  the  case  of  the  beam  by  one  third.  This 
renders  the  moment  curve  very  flat  at  the  center  and  the  moment 
increment  at  and  for  a  considerable  distance  each  side  of  the  diagonal 
center  very  small,  and  on  the  other  hand  it  renders  the  moment 
curve  correspondingly  sharper  toward  the  support,  and  hence 
effective  lateral  action  of  indirect  stress  must  occur  toward  the 
the  outer  edge  of  the  plate  since  the  intensity  of  bond  shear  depends 


LOCUS    OF    EFFECTIVE    ACTION    OF    INDIRECT    STRESS  135 

on  the  moment  increment.  The  moment  increment  being  nearly 
negligible  for  a  considerable  distance  about  the  diagonal  center  and 
zero  at  the  center,  the  efficiency  of  a  narrow  belt  is  negligible  from 
the  standpoint  of  plate  action  as  will  be  shown  experimentally  in 
tests  to  be  discussed  later  of  the  Mushroom  and  Norcross  type  of 
slabs. 

In  the  cantilever  area  about  the  column  the  following  significant 
conditions  are  to  be  considered : 

The  critical  section  is  at  the  support  and  the  resisting  section 
decreases  toward  the  support,  hence  the  necessary  radial  res:stance 
at  the  support  should  be  reduced  so  far  as  possible  by  circumferential 
resistance  as  the  support  is  approached  from  the  line  of  inflection. 

Fortunately  the  shear  at  the  line  of  inflection  is  about  esventy- 
five  percent  of  that  at  the  support  and,  as  the  increment  of  moment 


Fig.  49. 

depends  on  the  vertical  shear,  effective  action  of  indirect  stress  up 
to  the  line  of  inflection  is  secured  by  the  use  of  wide  belts  of  rein- 
forcement. 

Thus  the  arrangement  of  the  reinforcing  steel  laterally  as  well  as 
vertically  vitally  affects  the  characteristics  of  the  structure  in  such 
wise  that  when  the  belts  of  rods  which  run  in  multiple  directions  from 
support  to  support  are  spread  out  sufficiently  to  make  the  metal 
cover  the  entire  area  with  crossed  reinforcement  permitting  coaction 
of  bond  stresses  it  imparts  the  property  of  plate  action  to  the  slab 
in  which  circumferential  resistance  occurs  in  wide  areas  about  the 
columns  where  the  metal  is  at  the  top  and  also  in  the  central  area 
of  each  panel  where  the  metal  is  at  he  bottom. 

Without  such  wide  spreading  belts  full  and  effective  plate  action 
is  impossible.  Narrow  belts,  or  belts  part  of  them  in  the  bottom 
of  the  slab  at  the  columns,  do  not  realize  results  that  compare 
favorably  with  the  success  of  the  Mushroom  structure. 

2.  Slab  as  a  Mechanism.  Treating  the  Mushroom  slab  as  a 
mechanism  to  carry  the  load  to  the  supports,  the  result  of  its  opera- 
tion must  be  measured  and  compared  with  other  structures  in  terms: 


136 


APPLICATION    OF    CLAYPERON's    THEOREM    TO    THE    FLAT    SLAB 


First,  of  the  amount  of  load  it  can  carry  with  a  given  quantity 
of  material : 

Second,  in  terms  of  its  deflection  or  stiffness  under  the  given  load 
with  a  given  amount  of  steel  and  a  given  depth  of  slab. 

Clayperon's  theorem  that  "The  exterior  force  applied  multiplied 
by  the  displacement  in  the  direction  of  its  point  of  application 


Fig.  50. 

equals  the  sum  of  all  the  internal  work  of  a  body  elastically  deformed" 
gives  a  basis  for  ascertaining  the  manner  of  the  storage  of  potential 
energy  in  a  reinforced  structure  by  which  it  is  possible  to  demon- 
strate the  difference  between  the  circumferential  cantilever  action 
in  a  Mushroom  slab  with  four  way  belts,  Type  IV,  and  linear  can- 
tilever action  such  as  1  hat  of  the  Hennebique  beam  type.  In  making 
this  comparison  let  the  same  thickness  of  slab  and  cross  section  of 
steel  be  assumed  in  the  two  cases  and  equal  column  spacing  in  both 
directions. 


RADIAL    AND    CIRCULAR    RESILIKXCK  137 

Take  a  circular  arc  of  radius  R  about  a  column  as  a  center,  then 
in  case  it  has  a  radial  deformation  AR  it  will  have  a  circumferential 
deformation  2wAR  which  makes  these  deformations  equal  per 
linear  unit.  But  since  the  steel  runs  in  multiple  directions  in  the 
belts  of  the  Mushroom  slab  with  substantially  uniform  spacing, 
equal  deformations  of  equal  reinforcement  radially  and  circum- 
ferentially  requires  the  same  work  of  deformation.  Hence  the  work 
stored  circumferentially  and  radially  is  substantially  the  same. 
This  proof  assumes  the  same  unit  deformation  at  all  distances  from 
the  center  such  as  would  occur  in  stretching  a  perfectly  flat  sheet. 
It  will  therefore  be  proper  to  give  a  proof  by  integration  of  elements 
applicable  to  the  case  of  a  bent  slab  where  the  deformation  is  not 
necessarily  the  same  at  different  distances  r  from  the  center. 

Referring  now  to  Fig.  50,  the  position  of  the  slab  bent  down- 
ward about  the  column  is  shown  in  an  exaggerated  manner.  An 
elongation  of  the  top  surface  or  fiber  of  the  slab  is  noted  as  A.K, 
for  a  radius  R  and  an  elemental  sector  included  between  radii  making 
an  angle  with  each  other  d  6. 

Treating  the  reinforcement  as  equivalent  to  a  sheet  of  uniform 
thickness,  we  deduce  the  relation  between  radial  and  circumferen- 
tial unit  stresses  in  the  following  manner: 

Problem:  In  case  of  uniform  stretching  of  a  circular  plate  one 
unit  thick,  to  compute  the  radial  and  the  circumferential  work 
against  the  elastic  forces  having  a  final  intensity  of  /  per  square 
unit  of  cross  section. 

Resilience  =  (mean  unit  stress)  X  (cross  section)  X  (elongation) . 
Mean  unit  stress  during  deformation  =  J/. 

a.      Radial  Resilience:     By  definition  of  modulus  E, 
R:  AR  ::*:/,.'./-    *£* 

iP    fl      f) 

Mean  cross  section  of  elementary  sector  = 

Hence  the  radial  resilience  of  an  elementary  sector 

=    EAR   X—-°     X   A  R  =  I  E  ( A  R)2  d  6. 
2R  2 

Total  radial  resilience  of  circular  plate 

T7T       /      A  73  \  2 

jTj      (   / _\     t\j  /  *97T  *7T  /  \  9 

4  •'"  2 


138  DEFLECTIONS    DETERMINED    BY    RADIAL    DEFORMATIONS 

b.     Circumferential  Resilience  of  elementary    ring    of   radius    r. 

By  geometry  2  IT  r  :  2  TT  # ::  2  w  A  r  :  2  TT  A  #  ::  E  :  /,  /.  /=  ^  ^ 

/£ 

Hence  the  circumferential  resilience  of  elementary  ring 

=  E  AR  Xdr  X  2*  Ar,  but  Ar  =  r  A  R 
2R  R 

.'.  Total  circumferential  resilience  of  circular  plate 


.'.  Total  circumferential  resilience  =  total  radial  resilience. 

In  considering  the  geometry  of  the  convex  areas  over  the  head  of 
the  columns,  or  the  concave  areas  about  the  diagonal  center  of  the 
panels,  produced  in  a  slab  by  loading  it,  we  notice  that  the  deflec- 
tions in  such  areas  are  completely  determined  and  measured  when 
we  know  the  deflection  of  the  meridian  or  radial  curves  of  those  areas. 
The  deflection  of  these  meridian  curves  involve  certain  elongations 
in  the  reinforcement  in  radial  directions.  They  also  involve  lateral 
or  circumferential  elongations.  But  while  these  lateral  deformations 
accompany  the  radial  deformations,  and  are  connected  with  them, 
the  deflections  may  be  regarded  geometrically  as  wholly  independent 
of  them,  and  taken  as  dependent  only  on  the  radial  deformations. 

Now  the  energy  of  deformation  stored  in  these  areas  under  load 
has  been  shown  to  be  equally  divided  between  that  done  radially 
and  that  done  circumf erentially ;  or  that  done  longitudinally  and 
that  done  laterally.  In  beam  action  all  the  energy  is  stored  longi- 
tudinally, and  we  proceed  to  make  a  comparison  of  this  kind  of  action 
with  the  slab  action  which  we  have  been  considering,  where  it  is 
half  longitudinal  and  half  lateral. 

Suppose  a  given  amount  of  energy  Q  is  stored  in  a  wide  portion 
of  a  slab  extending,  let  us  say,  diagonally  from  column  to  column, 
and  compare  its  deportment  under  load,  by  beam  action,  and  by 
slab  action.  According  to  beam  action,  this  energy  Q  is  due  to  a 
certain  load  Wi,  gradually  applied  to  the  beam,  which  has  a  mean 
deflection  of  hi,  say,  so  that  Q  =  ^Wihi. 

But  if  the  same  amount  of  energy  be  stored  in  this  area  regarded 
as  a  slab,  only  half  of  it  will  be  stored  longitudinally  so  that  the  work 
thus  stored  will  be  ^Q  =  ^W2h2,  half  of  it  in  the  steel  and  half  in  the 
concrete. 

But  with  half  as  much  energy  stored  longitudinally,  the  stress 


STIFFNESS    OF    BEAM    AND    SLAB    TYPES    COMPARED  139 

in  the  longitudinal  rods  will  be  one  fourth  that  in  regular  beam  action, 
and  the  deflection  also  one  fourth  as  great,  or  h-2  =  \hi. 

But  if  the  deflection  in  the  slab  is  only  one  fourth  as  great 
under  the  same  load  as  on  the  beam,  the  load  on  the  slab  would  have 
to  be  four  times  the  load  on  the  beam  in  order  to  produce  the  same 
deflection  in  both. 

3.  Relative  Stiffness  of  Beam  and  Slab.     The  significance  of 
this  result  for  slab  theory  may  be  set  in  a  clear  light  by  taking 
account  of  the  fact  that  a  continuous  beam  is  five  times  as  stiff  as  a 
simple  beam,  but  a  continuous  slab  is  four  times  as  stiff  as  a  con- 
tinuous beam,  consequently  a  continuous  slab  supported  on  points 
should  be  twenty  times  as  stiff  as  a  simple  beam  of  the  same  cross 
section  on  knife  edge  supports.     But  since  it  is  not  practicable  to 
support  a  slab  on  points  we  must  take  into  consideration  the  diameter 
of  the  usual  bearing  provided  therefor. 

Taking  the  breadth  of  the  bearing  into  consideration  would 
have  a  tendency  to  increase  the  relative  stiffness  of  the  plate  to  more 
than  twenty  times  that  of  the  continuous  beam  or  plate  supported 
on  parallel  supports;  but  this  will  be  referred  to  in  due  order. 

Now  consider  further  the  circumferential  action,  which  has  such 
a  remarkable  effect  upon  the  deflection  as  that  which  has  just  been 
deduced.  In  this  treatment  of  the  circumferential  frames  or  the 
belts  of  reinforcement,  it  was  assumed  that  the  rods  were  the  equi- 
valent of  a  sheet  of  metal  of  corresponding  section.  The  actual 
fact,  however,  is  that  they  approach  that  when  bound  together  in 
the  concrete  matrix.  Taking  a  diametral  section  across  a  column 
head  it  appears  that  the  pull  in  the  rods  across  the  section  cut,  es- 
pecially in  the  portion  of  the  belt  inside  the  cap,  is  in  position  to  resist 
bending  in  the  direction  of  these  rods,  and  hence  the  rods  forming 
the  frames  outside  the  cap  act  largely  in  virtue  of  the  bond  values 
of  the  concrete  in  which  they  are  embedded,  to  resist  stresses  in  the 
same  direction  as  the  radial  rods  parallel  to  them  inside  the  cap. 

This  is  confirmatory  of  the  relation  which  is  brought  out  by 
comparing  the  reinforcement  to  an  equivalent  sheet  of  uniform  cross 
section. 

4.  Indirect  Tension.     The  law  of  the  combination  of  concrete 
and  metal  to  resist  flexure  may  be  stated  in  the  following  terms: 

Every  pound  of  tension  or  stress  thruout  the  area  of  the  cross 
section  of  the  reinforcement,  is  induced  in  the  bar  by  the  indirect 


140  LAWS    OF    INDIRECT    TENSION 

tension  or  bond  value  of  the  adhesion  of  the  concrete  to  the  bar, 
which  causes  the  rod  to  coact,  or  work  with  the  concrete  matrix. 

Thus  it  appears  that  without  calling  for  any  resistance  to  direct 
tensile  stress  in  the  concrete,  the  embedment  of  the  steel  has  induced 
indirect  tensions  in  the  concrete  matrix  whose  total  amount  is  equal 
in  magnitude  to  the  tensile  stress  in  the  bars  of  the  circumferential 
frames.  This  indirect  tension  is  developed  by  the  shearing  bond 
between  the  bar  and  the  matrix,  and  these  stresses  are  in  reality  shears 
extending  along  the  surface  of  the  bar,  equivalent  to  indirect  tensions 
and  compressions  at  45°  thereto.  These  indirect  tensions  may  be 
thought  of  as  lines  of  force  radiating  from  the  surface  of  the  bar 
thru  the  mass  of  the  concrete,  and  engaging  in  action  with  and 
held  in  equilibrium  by  similar  forces  generated  at  the  bond  surface 
of  some  other  system  of  bars  reinforcing  this  zone  of  the  slab  at  an 
angle  to  the  first  system.  This  indirect  tensile  strength  differs  from 
direct  tensile  resistance  in  a  most  important  particular,  viz:  that  a 
check  or  crack  in  the  concrete  does  not  seriously  interfere  with  its 
continued  action. 

It  can  only  be  destroyed  by  the  complete  disintegration  of  the 
matrix,  and  thus  it  differs  from  the  direct  tensile  strength  of  concrete 
in  the  important  particular  that  it  is  dependable  and  permanent. 

The  preceding  explanation  may  not  appear  to  be  clearly  estab- 
lished without  a  comparison  of  the  action  of  these  indirect  tensile 
stresses  with  those  occurring  in  the  case  of  the  slab  reinforced  two 
ways  and  bent  under  load  in  one  direction  only,  as  a  simple  beam. 
In  a  test  of  a  two  way  slab  made  for  the  purpose,  and  as  the  result 
of  the  test  by  simple  bending  on  end  supports  it  was  found  that  no 
appreciable  interaction  of  indirect  tension  occurred  between  the 
longitudinal  bars  and  lateral  bars.  The  reason  for  this  becomes 
evident  when  we  consider  the  manner  of  the  distribution  of  the  lines 
of  force  in  the  mass  of  the  matrix.  As  they  leave  the  surface  of  the 
bar  these  lines  of  force  are  disseminated  thru  the  mass,  and  can 
only  be  engaged  in  action  and  held  in  equilibrium  by  similar  lines 
of  force  emanating  from  the  surface  of  another  bar  or  bars  also  under 
tension  at  an  angle  thereto,  for  the  reason  that  as  these  forces  pro- 
ceed from  the  surface  of  the  bar  thru  the  mass,  their  intensity 
decreases  inversely  as  the  square  of  the  distance,  following  the  usual 
law  of  transmission  of  force  thru  mass.  Thus  the  area  of  the 
surface  of  a  transverse  bar  not  under  stress  is  relatively  too  small 
for  these  forces  to  act  upon  and  coact  with,  when  the  bar  is  not 
under  tension,  since  it  does  not  generate  similar  forces  which  can 


DIFFERENCE    BETWEEN    DIRECT    AND    INDIRECT    TENSION    IN    CONCRETE       141 

coact  with  the  first  set,  except  when  it  is  itself  under  stress.  It  is 
further  evident  that  otherwise  there  is  no  reason  for  these  forces 
to  converge,  but  every  reason  for  them  to  continue  to  diverge  as  they 
proceed  from  the  generating  surface. 

The  conditions  existing  in  the  concrete  and  steel  of  a  simple  beam 
under  small  loads,  amply  justifies  this  conclusion.  The  indirect 
tensile  stresses  emanating  from  the  surface  of  the  rods  finds  no  other 
resistance  than  the  direct  tensile  strength  of  the  concrete  to  react 
against.  Consequently,  as  the  concrete  itself  is  also  elongated  in 
tension,  cracks  occur  early,  leaving  these  indirect  tensions  to  coact 
with,  or  react  upon  separated  sections  of  the  concrete  relieved  from 
the  action  of  direct  tensions.  The  only  function  of  the  concrete 
in  the  tension  zone  after  these  tension  cracks  occur,  is  to  transmit 
shears  horizontally  and  vertically,  and  it  can  store  up  no  other  po- 
tential energy  except  that  of  shear.  The  fact  that  the  beam  acts 
as  a  beam  at  all  shows  that  the  horizontal  shearing  stresses  essential 
to  beam  action  are  in  full  operation.  It  is  thus  that  the  simple 
beam  retains  its  load  resisting  capacity  under  steel  stresses  far  above 
those  under  which  the  first  cracks  in  the  concrete  appear. 

This  change  of  function  of  the  concrete,  differentiates  sharply 
between  direct  tensile  stress  in  the  concrete,  which  may  not  be  relied 
upon  as  an  element  of  strength,  and  indirect  stresses,  which  neces- 
sarily exist  in  all  reinforced  beams  and  slabs  up  to  the  point  of  com- 
plete disintegration  of  the  concrete.  Indirect  stresses  are  the 
necessary  basis  of  beam  action  in  all  cases,  and  are  depended  on  by 
all  constructors. 

For  example,  it  has  been  shown  in  numerous  tests  that  a  simple 
beam  can  be  loaded  to  a  point  where  the  computed  stress  on  the  steel, 
disregarding  the  concrete,  would  be  approximately  nine  thousand 
pounds  to  the  square  inch,  with  measured  elongations  in  the  steel 
representing  only  five  thousand  pounds  per  square  inch.  But  after 
the  concrete  cracks  under  the  direct  tension,  the  stress  in  the  steel 
rapidly  rises  to  the  amount  computed  by  the  usual  formula,  and  the 
beam  may  be  further  loaded  until  the  yield  point  value  of  the  steel 
is  developed. 

It  thus  appears  that  the  useful  energy  stored  up  in  one  way 
reinforcement  by  indirect  tension  is  limited  by  the  fact  that  these 
tensions  must  react  against,  and  be  held  in  equilibrium  by  the 
direct  tensile  reistance  of  the  concrete;  whereas,  in  multiple  way 
reinforcement,  on  the  other  hand,  these  tensions  react  upon  similar 


142 


MEASURED    AND    COMPUTED    MOMENTS    IN    BEAMS    COMPARED 


indirect  tensions  disseminated  through  the   concrete   and  induced 
by  one  or  more  systems  of  reinforcement  inclined  thereto. 

During  the  period  that  the  tensile  resistance  of  the  concrete  in  a 
beam  is  in  full  operation,  or,  in  other  words,  its  modulus  of  elasticity 
in  tension  remains  constant,  as  much  potential  energy  is  stored  up 
by  indirect  tensions  in  the  concrete  as  is  stored  up  within  the  steel, 
and  also  an  additional  amount  by  direct  tension  in  the  concrete. 

Hence  during  this  period  the  measured  stress  upon  the  steel  is 
less  than  half  that  computed  according  to  the  usual  beam  formula 
by  disregarding  the  tensile  stress  on  the  concrete. 

We  may  illustrate  this  point  by  reference  to  Bulletin  No.  28, 
University  of  Illinois,  by  Talbot,  Table  3,  page  16,  describing  tests 
of  large  reinforced  concrete  beams. 

TABLE 


Applied 

Deflection 

Stress  in  Steel  in  Pounds  Per  Square  Inch 

Load 

in 

in 

Inches 

From  Deforma- 

From Bending 

Pounds 

tion 

Moment 

67,000 

.01 

300 

4,100 

103,000 

.02 

1,200 

6,300 

131,000 

.03 

2,100 

8,000 

159,000 

.05 

3,600 

9,700 

189,000 

.10 

6,000 

11,500 

220,000 

.11 

7,500 

13,400 

253,000 

.13 

10,500 

15,300 

285,000 

.17 

14,700 

17,400 

350,000 

.24 

20,100 

21,400 

414,000 

.31 

25,800 

25,300 

The  working  load  for  above  beam  was  290,000  Ibs.  impact 
included;  so  that  for  the  working  load  proper  no  considerable  energy 
of  tensile  resistance  is  stored  in  the  concrete.  In  other  words,  the 
storage  reservoir  of  energy  in  the  concrete  of  a  one  way  reinforced 
slab  or  beam  is  a  leaky  one  which,  because  it  involves  stress  in  ten- 
sion upon  the  concrete  approaching  the  ultimate  strength  of  the 
concrete,  allows  the  stored  energy  to  escape  or  be  dissipated  by  the 
permanent  yielding  of  the  concrete  or  its  final  cracking;  in  either 


STIFFNESS    OF    SLAB    AND    BEAM    TYPES    COMPARED  143 

event,  unloading  the  stress  upon  the  steel  which,  because  the  steel 
unlike  the  concrete  not  being  in  a  condition  of  overstrain,  is  the  more 
rigid  element.  This  result,  the  shifting  of  work  from  concrete  to 
steel,  is  accomplished  either  by  semi-plastic  local  stretching  in  the 
matrix,  or  by  actual  cracking. 

This  test  by  Talbot  lacks  an  interesting  feature  brought  out  by 
tests  we  have  recently  had  made,  and  that  is  this:  that  if  a  given 
load  be  left  upon  the  beam  for  a  long  time  there  is  a  leaking  away 
of  the  energy  stored  in  the  concrete,  even  with  a  perfectly  quiescent 
load  of  low  intensity;  since  concrete  cannot  endure  a  stress  much 
exceeding  twenty-five  percent  of  its  ultimate  direct  tensile  strength 
without  some  increase  in  strain  with  no  change  in  the  applied  load. 

This  test  beam  of  Professor  Talbot  was  25' 0"  out  to  out  and  2'  10" 
deep,  a  larger  ratio  of  depth  to  span  than  is  usual,  and  the  effects 
above  noted  are  less  marked  than  would  be  the  case  with  a  beam  hav- 
ing a  smaller  depth  relative  to  length  of  span. 

The  values  observed  for  the  three  lower  loads  indicate  the  mask- 
ing effect  of  shrinkage  stress  in  curing  rather  than  true  elastic  de-. 
portment. 

Returning  now  to  the  comparison  of  slab  and  beam  action, 
the  difference  in  mechanical  operation  is  such  that  the  continuous 
flat  slab,  Mushroom  type,  requires  the  same  amount  of  steel  to  carry 
a  given  load  as  a  beam  construction  having  beams  of  a  depth  equal 
to  twice  that  of  the  slab.  The  slab  construction  has  the  further 
advantage  over  the  beam  in  that  its  reinforcement  covers  the  area 
of  the  floor  while  additional  slab  reinforcement  is  requisite  to  com- 
plete the  beam  and  slab  floor.  This  relation  gives  the  slab  type 
its  greatest  advantage  for  heavy  loads,  tho  its  utility  is  not  limited 
to  heavy  construction  alone. 

The  deflections  compared  have  been  limited  to  that  of  a  simple 
beam  supported  on  knife  edge  supports  with  that  of  the  continuous 
slab  supported  on  points.  The  actual  support  must,  of  course, 
have  considerable  breadth,  usually  about  0.22  of  the  span  for 
ordinary  slabs.  Such  supports  would  reduce  the  net  span  to  about 
0.8  of  the  distance  center  to  center  of  columns  directly  and  to  1.214 
times  the  direct  span  in  a  diagonal  direction,  the  diagonal  of  the 
square  being  1.414  L  less  .2L,  the  net  diameter  of  the  cap. 

Accordingly,  on  this  basis  the  stiffness  of  the  diagonal  as  compared 
with  that  of  the  simple  beam  on  knife  edge  supports,  having  the  same 
span  as  the  diagonal  of  the  panel,  should  be  inversely  as  the  cubes 


144  MID    SPAN    DEFLECTIONS    OF    DIRECT    AND    DIAGONAL    BELTS 

of  the  ratio  of  the  spans.  But  it  has  already  been  shown  that  a 
slab  supported  on  points  at  the  corners  is  theoretically  twenty  times 
as  stiff  as  a  simple  beam  of  span  L.  But  the  cap  increases  this 
stiffness  in  the  ratio  of  1.4143  to  1.2143  or  1.58  to  1.  But  1.58X20 
=  31.6,  hence  the  cap  would  increase  the  stiffness  to  nearly  32  times 
that  of  a  simple  beam  of  span  L,  were  it  not  for  the  disparity  in  the 
resisting  section  as  the  support  is  approached  in  case  of  the  con- 
tinuous slab  as  compared  with  the  slab  of  constant  section.  The 
critical  section  for  bending  is  at  the  support  or  around  it  where  the 
negative  bending  moment  is  greatest  and  where  the  resisting  material 
presents  not  a  constant  section  as  regards  the  concrete  but  one  re- 
ducing or  growing  smaller  as  the  support  is  approached.  With  an 
effective  cap  0.2  of  the  span  in  diameter,  the  resisting  section  is 
approximately  0.2-7rL  =  0.63L  in  circumference  and  the  resisting  sec- 
tion at  the  line  of  inflection  taken  roughly  as  circular  is  about  1.4 
times  the  span  in  circumference  which  combined  with  the  massing 
of  the  steel  at  the  support  would  require  but  a  slight  reduction  in 
this  ratio  of  32  to  perhaps  27  in  order  to  arrive  at  the  true  deflection. 

A  continuous  slab,  such  as  one  of  the  Mushroom  type,  is  not  in 
a  strict  sense  an  imitation  of  a  homogeneous  plate  because  the  steel 
percentage  varies  through  wide  limits  in  different  parts  of  the 
slab.  It  approaches  1J  or  If  per  cent  at  and  close  to  the  column, 
which  is  permissible  on  account  of  the  manner  in  which  the  concrete 
is  strained  under  compression  at  the  bottom  as  the  support  is  ap- 
proached, while  it  is  reduced  in  the  thin  slab  to  the  limit  of  .2  percent 
at  mid  span  in  the  side  belts  where  there  is  one  layer  only  of  rods 
and  to  .4  percent  at  the  diagonal  center,  so  that  its  moment  of 
resistance  as  regards  reinforcement  is  varied  thru  wide  limits 
in  order  to  proportion  the  steel  to  the  total  stress  in  the  various 
parts  of  the  panel  as  required  by  slab  action,  and  hence  such  a  slab 
is  more  scientific  and  economical  than  a  mere  imitation  in  the  form 
of  a  homogeneous  plate  of  uniform  thickness.  This  difference  is 
comparable  to  the  difference  between  a  truss  and  a  beam  of  constant 
section,  a  truss  providing  the  most  material  where  it  is  most  needed, 
while  the  beam  of  constant  section  is  wasteful  of  material  in  that 
it  is  not  proportioned  with  reference  to  the  stresses  brought  on  it 
by  the  applied  forces. 

5.  Comparison  of  Deflections  at  Mid  Span  of  the  Diagonal 
Belt  and  the  Direct  Belt.  Having  shown  the  manner  of  storage  of 
the  potential  energy  of  inlernal  work  in  the  cantilever  portion  of 
the  slab  and  in  the  suspended  circular  plate  about  the  diagonal  of 


DIVISION    OF    LOAD    IN    SHEAR  145 

the  panel  to  be  such  that  it  is  equally  divided  so  far  as  the  slab 
rods  are  concerned  between  radial  and  circumferential  deforma- 
tions, and  having  derived  an  approximate  value  for  the  deflection 
at  this  point  based  upon  these  relations,  and  upon  the  reduction  of 
span  by  the  diameter  of  the  cap,  it  is  in  order  to  compare  the  rela- 
tive magnitude  of  the  deflection  at  mid  span  of  the  direct  and 
diagonal  belts  respectively. 

Considerations  of  symmetry  and  equal  width  of  belt  for  square 
panels  with  the  wide  over-lap  of  these  belts  where  their  width  is 
7/16  to  1/2  the  length  of  span,  would  indicate  substantially  equal 
division  of  the  load  in  shear  resistance  between  the  four  respective 
belts  and  on  this  basis  we  may  proceed  to  compare  the  deflections 
at  mid  span  directly  and  diagonally  between  columns.  There  is  no 
plate  action  by  the  indirect  stress  at  the  center  of  the  direct  belt. 
The  resistance,  however,  of  the  direct  belt  is  largely  augmented  by 
the  over-lapping  of  the  diagonal  belts  when  they  are  of  sufficient 
width  to  intersect  within  or  approximately  at  the  edge  of  the  direct 
belt.  Their  assistance  to  the  direct  belt  may  be  considered  then  as 
the  addition  of  an  effective  section  which  may  be  without  material 
error  considered  as  in  proportion  to  the  extent  of  over-lap.  Thus 
with  J  L  for  the  width  of  belt  in  a  square  panel,  the  over-lapping 
of  the  diagonal  belts  may  be  considered  to  increase  the  resisting  sec- 
tion'of  the  direct  belt  by  approximately  .7  while  if  the  width  of  belt 
is  7  /1 6  L  the  increase  in  effective  section  may  be  considered  as  .5 
the  section  of  the  direct  belt. 

The  formula  for  stiffness  previously  derived  shows  that  the 
stiffness  for  the  same  load  is  inversely  as  the  cube  of  the  span  and 
inversely  as  the  cross  section  of  the  resisting  steel.  In  the  diagonal 
direction  plate  action  doubles  the  efficiency  of  the  steel.  Hence 
we  may  compare  the  deflections  of  the  direct  and  diagonal  belts 
on  the  basis  of  the  cube  of  the  diagonal  span  between  lines  of  in- 
flection, divided  by  the  cube  of  the  direct  span  between  lines  of 
inflection  over  the  effective  steel  sections  for  the  respective  belts. 
The  effective  section  for  the  direct  belt  as  we  have  pointed  out  being 
from  1.5  to  1.7  of  the  section  of  the  direct  belt,  these  ratios  are 
reduced  approximately  or  roughly  to  that  of  the  ratios  of  the  squares 
of  the  spans  in  the  direct  and  diagonal  directions.  This  gives  us 
as  a  rough  computation  a  deflection  at  the  diagonal  center  about 
1.4  times  that  at  mid  span  of  the  direct  belts  where  the  diameter  of 
the  cap  =  .2L  and  all  panels  are  loaded  and  the  amount  of  these 
deflections  we  can  figure  from  the  ratios  previously  derived  and  com- 


146  APPARENT    BENDING    MOMENTS 

pare  with  the  computed  deflection  of  the  simple  beam  of  constant 
section.  The  deflection  at  mid  span  of  a  direct  belt  is  not  the  same 
when  a  single  panel  only  is  loaded,  tho  that  at  the  panel  center  is 
unchanged. 

6.  Bending  Moments.  The  moment  of  the  applied  forces  in 
the  case  of  a  uniform  load  acting  on  an  interior  panel  of  a  continuous 
four  way  slab  assumed  to  be  supported  on  columns  uniformly  spaced 
is  determined  as  follows: 

From  the  fundamental  relations  of  moment  magnitudes  we  know 
that  for  uniform  loads  half  the  sum  of  the  moments  over  the  supports, 
plus  that  at  mid  span,  equals  a  constant,  WL  /8.  The  division 
of  this  moment  between  the  supports  and  mid  span  depends  on  the 
nature  of  the  design  in  respect  to  the  relative  rigidity  of  the  con- 
struction at  the  support  and  mid  span  respectively. 

The  relative  rigidities  of  these  sections,  (presupposing  of  course, 
moment  resisting  capacity  in  the  slab  thruout)  fixes  the  position  of 
the  line  of  inflection  which  divides  the  cantilever  portion  from  the 
suspended  span  portion  of  the  construction. 

If  now  these  relative  rigidities  are  made  such  that  the  position 
of  the  line  of  inflection  is  located  the  same  distance  from  the  support 
as  in  a  beam  fixed  at  its  ends,  then  the  apparent  moment  over  the 
support  is  WL'  /12  and  TFL'/24  at  mid  span.  But  if  L  denoting 
the  distance  center  to  center  of  columns  rather  than  the  distance  L' 
center  to  center  of  supports  near  the  edge  of  the  cap,  is  to  be  used 
in  the  above  expression  then  these  coefficients  must  be  reduced  in 
like  ratio  to  the  reduction  of  span  by  the  cap  diameter  in  order  to 
substitute  L  for  L'  in  the  moment  values.  Thus  for  an  effective 
cap  of  .2L  these  moments  in  terms  of  L,  (the  distance  center  to 
center  of  column)  become  W  L/15  over  the  support  and  TFL/30 
at  mid  span,  which  values  are  to  be  modified  in  like  manner  for 
other  values  of  cap  diameter. 

It  has  been  shown  that  the  manner  of  storage  of  the  potential 
energy  of  internal  work  in  the  cantilever  portion  of  the  slab  is  such 
that  it  is  to  be  equally  divided  so  far  as  the  slab  rods  are  concerned 
between  the  radial  and  circumferential  deformations,  and  this  method 
of  distribution  would  require,  under  the  above  proportions  of  cap 
diameter  to  span  that  provision  be  made  to  resist  a  moment  in  a 
radial  direction  of  W  L/30  at  the  edge  of  the  cap. 

In  proportioning  the  steel,  the  critical  section  for  bending  should 
be  taken  as  a  circle  about  the  capital  and  of  a  diameter  for  the 


TRUE    MOMENTS    OF    STEEL    STRESS  147 

effective  cap  on  which  the  steel  rests,  of  the  diameter  of  the  cap  plus 
If  times  the  slab  thickness  minus  4  to  6  inches,  depending  on  the 
size  of  the  panels. 

The  consideration  of  symmetry  and  equal  width  of  belt  for  square 
panels  would  indicate  a  substantially  equal  division  of  the  total 
load  in  the  four  belts  respectively,  and  on  this  basis  it  is  possible 
to  determine  the  resisting  moment  required  of  the  steel  at  mid  span 
of  the  diagonal  belt  and  the  direct  belt. 

On  the  basis  of  this  division,  the  apparent  moment  to  be  resisted 
by  each  of  the  belts  at  mid  span  is  WL  /1 20  and  the  true  moment  of 
stress  in  the  steel  times  its  lever  arm  is  to  be  determined  from  the 
apparent  moment  and  the  mechanics  of  indirect  stress  heretofore 
discussed  at  length.  At  the  center  of  the  diagonal  belts,  slab  action 
doubles  the  efficiency  of  the  true  moment  of  the  steel  stress.  Hence 
the  true  moment  of  steel  resistance  at  the  center  of  the  diagonal 
belts  is  WL  /240.  The  conditions  to  be  met  at  mid  span  of  the 
direct  belt  still  remain  to  be  considered. 

In  referring  to  the  diagram,  Fig.  16,  showing  the  plan  of  the 
reinforcement  of  this  type  of  construction,  it  is  observed  that  the 
diagonal  belts  over-lap  the  direct  belts  and  increase  by  this  over- 
lap the  efficiency  of  the  direct  belt  by  .5  to  .7,  depending  on  whether 
the  width  of  belt  is  7  /16  or  J  the  span  in  width.  If  the  efficiency 
of  the  direct  belt  be  increased  by  the  over-lap  of  the  diagonal  belt 
7,  then  the  moment  to  be  resisted  by  steel  at  mid  span  of  the  direct 
belt  is 

TFL/1.7X120,  orTFL/204 

Whereas  if  the  over-lap  is  less  and  increases  the  resisting  steel 
area  in  this  direction  only  .5  then  the  moment  at  mid  span  is 
TFL/1.5X120,  or  WL/18Q 

The  difference  in  assistance  between  the  manner  of  coaction  of 
one  diagonal  belt  with  another  and  the  coaction  of  the  diagonal 
belt  with  the  direct  belt  in  resisting  moment  at  mid  span  should  be 
specifically  noted.  The  coaction  of  one  diagonal  belt  with  another 
is  brought  about  by  the  coaction  of  the  steel  in  one  belt  under 
strain  with  the  steel  in  the  other  belt  under  similar  strain  through 
the  indirect  stress  generated  by  bond  shear  while  the  assistance 
rendered  the  direct  belt  by  the  over-lap  of  the  diagonal  belt  at 
mid  span  is  largely  the  assistance  rendered  by  the  addition  of 
greater  cross  sectional  area,  two  radically  different  methods  of 


148 


MEASURED    DEFLECTIONS    AND    STRESSES    COMPARED 


working  together  and  yet  each  adding  to  the  efficiency  of  the  com- 
bination structure. 

Having  explained  the  method  of  arriving  at  the  stress  at  mid 
span,  and  the  stress  over  the  support  for  uniform  load,  and  different 
diameters  of  cap,  it  is  next  in  order  to  discuss  the  effect  on  the  con- 
tinuous beam  of  loads  on  single  spans  as  contrasted  with  all  spans 
loaded  uniformly. 

In  a  continuous  beam  of  indefinite  length,  the  loading  of  alternate 
spans  has  a  very  marked  effect  upon  the  condition  of  stress  in  un- 
loaded spans.  The  top  of  the  beam  is  heavily  strained  in  tension 


£1 

o  s? 
!& 


1  pe 

le,l  s 


7 


,Lo 


/Loa 


Load  3 4512 


d  2  =361 


w-Load  5  =  1004 


4  -720 


TRESS  AND  DEFLECTION 
CURVES 

TEST  OF 

NORTHWESTERN  GLASS  CO., 
BUILDING' 
MINNEAPOLIS. 


|3  Defl.  scales,  1  space= 0.050  in.  defl.       1  jspace=0.150  jn.  defl.  1  epace=0.10  in.  defl. 

All  stress  scales,  1  space  =4000  Ib.  per  sq.  in. 

Fig.  51.     Showing  Agreement  of  Curves  of  Measured  Deflection  and  Measured 
Stress,  Mushroom  Floor  Slab. 

thruout  the  unloaded  span  and  the  bottom  flange  is  strained  in 
compression,  the  negative  moment  for  live  load  being  constant  in 
the  top  flange  thruout  the  unloaded  span  and  equal  to  the  moment 
at  the  support  if  the  dead  load  of  the  beam  itself  be  neglected. 

This  condition  of  strain  results  in  a  negative  deflection  at  mid 
span  of  the  unloaded  span  in  magnitude  equal  to  a  large  fraction 
of  the  deflection  in  the  loaded  span.  Accordingly  a  comparison  of 
the  positive  deflection  of  a  heavily  loaded  span  of  Mushroom 
construction  with  the  negative  deflection  in  adjacent  panels  will 
bring  out  and  show  in  strong  contrast  the  difference  in  the  me- 
chanics of  internal  stress  of  a  continuous  beam  and  a  continuous 
slab  supported  on  spaced  columns. 

The  principle  of  conservation  of  energy  fixes  rigidly  the  relation 
of  proportion  between  stress  and  deflection  at  mid  span  regardless 
of  the  method  of  reinforcement,  as  is  discussed  more  at  length  in 


COMPARISON    OF    CONTINUITY    IN    MUSHROOM    SLABS    AND    BEAMS        149 

the  comparative  test  of  the  Mushroom  and  Norcross  type  test  slab, 
Chapter  VI  and  also  illustrated  in  test  in  the  diagram  Fig.  51. 

In  the  test  of  the  Hoffman  Building,  Milwaukee,  Wis.,  under  a 
load  of  one  thousand  pounds  per  square  foot  of  sand  restrained  by 
sacks  in  the  outer  edge,  a  positive  deflection  of  7  /16  inch  was 
measured.  The  panel  was  16'  8"  by  17'  0",  the  basement  columns 
24"  in  diameter,  the  slab  8J"  thick.  Careful  measurements  were 
made  at  the  center  of  adjacent  panels  directly  and  diagonally  and 
no  appreciable  negative  deflection  could  be  observed. 

Considering  the  negative  moment  of  resistance  at  mid  span  in 
comparison  with  its  positive  moment  of  resistance,  it  is  obvious 
that  a  negative  moment  should  produce  a  negative  deflection  enor- 
mously larger  than  would  be  produced  by  an  equal  positive  moment, 
because  while  the  slab  is  well  reinforced  to  resist  positive  moment 
in  the  bottom  layer  at  mid  span,  there  is  no  steel  at  all  in  the  top 
at  mid  span  to  resist  negative  moment. 

Accordingly,  the  conclusion  is  unavoidable  in  view  of  the  test 
data  cited  and  many  similar  tests,  that  the  effect  of  loading  one  span 
is  almost  negligible  upon  adjacent  spans;  and  that  Mushroom  slabs 
must  be  consistently  treated  as  acting  in  a  large  measure  independ- 
ently of  each  other  as  regards  the  transmission  of  negative  bending 
moment  across  column  heads  and  sides  into  unloaded  panels,  thus 
differing  radically  from  a  continuous  beam  construction.  Such 
tests  do  not,  however,  show  that  a  considerable  amount  of  cir- 
cumferential and  radial  stress  is  not  transmitted  through  the  por- 
tion of  the  slab  around  the  head.  This  deportment  of  the  Mush- 
room slab  permits  the  consideration  of  any  panel  as  operating  to 
a  large  extent  independently  of  adjacent  panels.  In  other  words, 
panels  connected  integrally  with  the  columns  are  far  more  nearly 
self  contained  than  are  panels  cast  integrally  with  T-beams. 

In  the  treatment  of  beams  attention  was  called  to  the  fact  that 
the  continuous  beam  cast  integrally  with  the  column,  acts  more 
nearly  as  a  restrained  beam  than  a  continuous  beam  on  knife  edge 
supports.  So  the  Mushroom  panel  acts  in  a  larger  measure  as  a 
panel  restrained  at  the  columns  than  as  one  truly  continuous,  per- 
mitting the  design  of  all  panels  with  much  less  attention  to  the 
difference  in  continuity  of  the  slab  than  would  otherwise  be  per- 
missible. 

Careful  measurements  of  deflection  show  that  when  the  concrete 
work  is  old,  and  thoroly  cured,  little  difference  in  the  elastic  deport- 
ment is  to  be  noticed  between  an  end  panel  and  an  interior  panel. 


150  ONE-WAY    SLABS    AND    SLABS    OF    TYPE    III    FLOORS    COMPARED 

On  the  other  hand,  before  the  concrete  has  become  thoroly  dry, 
hard  and  rigid,  and  in  all  cases  where  the  forms  are  prematurely 
removed,  the  rigidity  of  the  end  panel  is  much  less  than  that  of  an 
interior  panel  and  special  care  should  be  exercised  in  conservatively 
supporting  wall  panels  until  the  concrete  is  thoroly  rigid  and  hard. 
This  consideration  has  led  to  the  practice  of  increasing  the  steel  in 
the  direct  belts  of  wall  panels  longer  than  eighteen  feet  where  half 
Mushroom  heads  are  used  next  to  the  wall  by  ten  percent,  and 
where  the  wall  end  of  the  slab  is  framed  into  a  beam  or  a  wall  an 
increase  in  the  direct  belt  of  fifteen  percent  is  made  standard  practice. 
No  increase,  however,  is  needed  in  the  diagonal  belts  since  the  object 
is  to  secure  such  a  reduction  of  the  deflections  of  those  side  belts 
which  are  perpendicular  to  the  wall  as  to  reduce  the  trough  of  the 
panel  which  is  parallel  to  the  wall  in  somewhat  the  same  ratio  that 
the  trough  perpendicular  to  the  wall  has  been  reduced  by  the  wall 
itself,  thus  insuring  ordinary  plate  or  slab  action  in  these  wall  panels 
by  the  approximate  equality  of  the  troughs  crossing  each  other 
in  them. 

The  rigidity  of  the  columns  affects  the  deportment  of  both 
the  continuous  beam  and  the  continuous  slab,  as  we  have  hereto- 
fore pointed  out.  In  a  continuous  slab,  supported  by  columns 
continuing  upward  for  one  or  more  stories  above,  the  amount  of 
restraint  is  evidently  greater  than  can  be  secured  in  a  floor  which 
is  merely  supported  by  columns,  since  in  the  latter  case  the  bending 
resistance  of  the  columns  is  reduced  fully  fifty  percent.  Accord- 
ingly, in  the  case  of  slabs  thus  supported  it  has  been  made  standard 
practice  to  increase  the  steel  in  the  direct  belts  of  such  construction 
by  ten  percent,  making  no  increase  whatever  in  the  diagonal  belts, 
with  a  corresponding  addition  in  wall  panels  dependent  on  the 
manner  in  which  they  are  supported,  whether  they  are  integral  with 
beams  or  with  half  Mushroom  heads. 

7.  Continuity  in  Flat  Slabs  and  in  Thin  Slab  on  Beams,  Con= 
trasted  Experimentally.  In  a  one-way  slab  on  continuous  beams 
freely  supported,  the  negative  moment  at  the  support  in  the  case  of 
alternately  loaded  panels  is  transferred  across  the  column  to  the  un- 
loaded panel  with  no  dimunition  except  that  due  to  the  positive 
moment  of  the  dead  load.  Where  the  continuous  beam  is  of  concrete 
and  rigidly  built  into  the  column  this  negative  moment  transferred 
across  to  the  unloaded  span  is  not  only  reduced  by  the  dead  load 
of  the  span  not  covered  by  live  load,  but  is  also  reduced  by  a 
moment  due  to  the  stiffness  and  rigidity  of  the  columns  into  which 


UNBALANCED    MOMENTS    NOT    TRANSFERRED    FROM    PANEL    TO    PANEL         151 

the  beam  is  framed  and  acts  as  a  monolith.  This  reduction  in 
the  negative  moment  depends  for  its  amount  evidently  on  the 
relative  rigidity  of  the  columns  and  beams. 

In  the  case  of  the  heavy  warehouse,  where  the  columns  are  large 
and  stiff,  the  negative  moment  transferred  to  mid  span  of  the  un- 
loaded panel  would  be  relatively  small  but  where  the  beam  is  framed 
into  a  girder  the  only  resistance  offered  to  reduce  the  transfer  of  this 
moment  at  mid  span  of  the  panel  is  the  twisting  resistance  offered 
by  the  girder,  which  is  relatively  small.  Hence  large  negative 
moments  may  be  transferred  from  the  loaded  to  the  unloaded  panel 
in  beam  construction  where  the  columns  are  light  or  the  continuous 
beam  frames  into  girders.  The  modifying  effect  of  these  supports 
upon  the  distribution  of  moments  in  continuous  beams  is  in  strong 
contrast  with  that  which  occurs  in  continuous  slab  floors  of  either 
of  the  natural  types  of  reinforced  concrete  III  or  IV. 

In  the  test  of  the  Hoffman  Building  in  Milwaukee,  a  load  of  1000 
pounds  per  foot  on  a  single  panel  16'8"  by  17'0",  or  a  total  load  of 
142  tons,  gave  a  deflection  at  the  diagonal  center  of  the  panel  of 
7  /16".  No  appreciable  negative  deflection,  however,  was  observed 
at  the  center  of  either  of  the  panels  adjacent  to  the  loaded  panel 
laterally  or  diagonally.  The  supporting  column  in  this  case  was  24" 
diameter,  reinforced  with  twelve  1-J  inch  rounds.  Story  height 
from  basement  to  the  first  floor  was  10'6."  The  second  tier  of 
columns  were  not  connected  to  the  basement  columns,  but  a  cast 
iron  base  was  provided  to  form  a  bearing  between  the  second  tier  and 
the  basement  tier.  The  panel  was  reinforced  with  seventeen  f " 
round  rods  in  four  directions  and  was  8J"  thick,  including  one  inch 
finish  coat. 

It  is  evident  that  columns  of  this  diameter  and  arranged  in 
this  manner  would  be  of  insufficient  rigidity  to  resist  so  great  a 
test  load  as  this  and  prevent  the  transference  of  moment  to  the 
adjacent  panel  if  acting  on  the  principle  of  the  beam.  Predominat- 
ing circumferential  action,  however,  prevents  the  linear  transference 
of  moments  in  the  continuous  flat  slab  in  the  same  manner  as  in  the 
continuous  beam,  and  this  is  the  reason  that  no  negative  deflections 
were  observed  at  the  centers  of  the  panels  adjacent  to  the  loaded 
panel  laterally  or  diagonally. 

Similar  action  occurs  in  square  panels  reinforced  in  two  directions 
and  supported  on  beams  running  from  column  to  column  in  both 
directions.  A  very  heavy  test  load  placed  upon  a  single  panel  has 
little  effect  upon  adjacent  panels  because  the  resistance  is  in  a  large 


152 


POSITION    OF    LINE    OF    INFLECTION 


measure    circumferential,    which    prevents    moments    from    being 
transferred  from  one  panel  to  another. 

Bearing  in  mind  that  circumferential  stresses  are  coincident 
with  and  dependent  for  their  magnitude  upon  radial  stress,  it  is 
evident  that  stresses  of  this  kind  cannot  be  transferred  from  panel 
to  panel  after  the  manner  of  stress  transferred  longitudinally  in  a 
member,  as  in  the  case  of  a  beam.  Numerous  tests  of  wall  panels 
and  interior  panels  in  two-way  beam  contruction  by  Turner  prove 
beyond  question  that  substantially  the  same  formula  applies  both 
to  an  interior  and  to  a  wall  panel  of  type  III,  even  where  this  wall 
panel  at  one  side  is  merely  built  into  the  brick  side  wall  of  the  build- 
ing. Further  that  a  corner  panel  built  into  the  brick  wall  in  the  same 
manner  exhibits  the  same  degree  of  elastic  resistance  to  deflection 
under  load  as  an  interior  panel  forming  one  of  a  continuous  monolith 
extending  for  a  number  of  panels  each  way  and  supported  on  four 
sides  by  beams  giving  the  same  clear  span  as  in  the  case  of  the  wall 
panel.  While  deflections  of  wall  panels  and  interior  panels  are  almost 
identical,  it  does  not  follow  that  there  is  no  difference  in  the  dis- 
tribution of  stress  in  the  two  cases  which  will  be  brought  out  more 
at  length  in  considering  the  deformations  which  occur  in  such  cases. 

8.  Variation  in  the  Position  of  the  Line  of  Inflection  in  Con= 
tinuous  Flat  Slab  Construction.  The  law  of  rigidities  previously 
discussed  fixes  the  position  of  the  line  of  inflection,  subject,  however, 
to  the  proviso  that  the  slab  be  able  to  resist  both  positive  and  nega- 
tive moments  in  the  zone  thru  which  this  change  takes  place.  But 
wide  variations  in  the  position  of  the  line  of  inflection  cannot  occur 
in  continuous  flat  plate  construction  because  its  position  is  fixed 
approximately  by  the  amount  of  metal  in  the  case  of  the  Mushroom 
system  and  by  the  dip  or  sharp  bending  down  of  the  slab  steel  in 
some  other  forms,  so  that  the  lines  of  inflection  are  near  the  points 
where  the  belts  cross  from  above  to  below  the  neutral  axis.  This 
limits  the  possible  variations  in  position  of  the  lines  of  inflection  to 
relatively  small  distances. 

However,  the  law  of  variation  of  the  position  of  the  lines  of  in- 
flection with  variations  in  the  moments  over  the  supports  and  at 
mid  span  follow  quite  a  different  law  in  continuous  flat  slabs  from 
what  it  follows  in  beams.  What  this  law  is  will  appear  by  con- 
sidering the  magnitudes  of  the  moments  as  expressed  in  terms  of 
the  load  w  on  unit  area  instead  of  expressing  it  in  terms  of  the  total 
load  W.  The  sum  of  half  the  moment  over  the  support,  plus  that 
at  mid  span  is  a  constant,  =  W  L  /8.  For  a  beam  of  breadth  b  this 


CONSIDERATIONS    DETERMINING    POSITION    OF    LINE    OF    INFLECTION  153 

is  w  b  L2  /8,  but  for  a  square  panel  it  is  w  L3/8.  This  addi- 
tional power  of  L  is  introduced  in  the  case  of  the  continuous  slab 
with  equal  column  spacing  in  both  directions  because  L  is  the 
breadth  of  the  panel,  so  that  its  load  per  foot  of  length  is  w  L,  and 
the  total  load  w  L2.  Thus  the  ratio  of  the  lengths  of  the  sus- 
pended spans  between  the  lines  of  inflection  will  vary  for  given 
applied  moments  as  the  cube  root  of  the  ratio  of  the  given  moments 
determined  by  the  relative  rigidities.  Thus  in  the  preceding  example, 
if  the  moment  at  mid  span  be  increased  twenty-three  percent,  since 
the  cube  root  of  1.23  is  1.07  there  would  be  a  corresponding  increase 
in  the  length  of  the  suspended  span  .577  L  of  approximately  seven 
percent  or  four  percent  of  the  span  L,  and  a  variation  in  position 
of  the  line  of  inflection  of  two  percent  of  the  span  L,  a  variation 
which  is  not  great,  and  one  which  in  view  of  the  fact  that  the  line  of 
inflection  is  not  clearly  defined  by  a  true  hinge  might  readily  occur. 

In  finding  the  moment  magnitudes  at  the  support  and  mid  span, 
we  have  made  use  of  the  fundamental  relation  of  moment  magnitudes 
and  assumed  that  the  lines  of  inflection  are  located  in  the  same 
position  as  in  fixed  beams  of  constant  section. 

In  the  early  constructions  of  Mushroom  floors,  the  rolling  mills 
were  not  prepared  to  furnish  small  rods  in  long  lengths  and  hence 
all  belts  of  rods,  as  a  rule,  were  lapped  or  spliced  over  the  column. 
With  such  an  arrangement  of  metal,  there  would  be  twice  the  sec- 
tion of  the  slab  steel  over  the  column  that  would  occur  with  long 
lengths  of  rods  and  no  splices  at  the  support.  This  difference  of 
one  hundred  percent  in  slab  steel  area  at  the  support  is  a  variation 
which  may  occur,  and  demands  investigation. 

If  we  over-reinforce  the  concrete  at  the  support,  the  concrete 
element  determines  to  a  large  extent  the  relative  rigidity  of  the 
cantilever  at  the  support.  The  shifting  of  the  neutral  plane  would 
be  by  no  means  in  proportion  to  the  increase  in  the  percent  of  steel 
as  may  be  observed  by  reference  to  the  k  curves  in  the  diagram  shown 
in  Fig.  25.  Accordingly  a  large  excess  of  steel  over  and  above  that 
necessary  would  add  a  relatively  small  amount  to  the  rigidity  of  the 
cantilever.  On  the  other  hand,  however,  the  cantilever  must  be 
so  designed  that  the  steel  provided  is  not  over-strained  therein. 

With  the  column  cap  approximately  0.2  of  the  span  length, 
the  applied  moment  of  W  L  /15  in  180°  at  its  edge  may  be  considered 
as  a  fair  average  value  and  the  steel  proportioned  accordingly,  for 
a  slab  of  uniform  thickness. 


154  FINISH    COAT    AND    EFFECT    ON    DEFLECTION 

9.  Effect  of  Adding  Finish  to  the  Rough  Slab.     A  finish  coat 
does  not  add  to  the  resistance  of  the  cantilever.     But  the   finish 
coat  adds  to  the  resistance  at  mid  span,  since  it  increases  the  effec- 
tive depth  of  the  slab  and  its  corresponding  rigidity,  and  it  is  now 
in  order  to  discuss  the  effect  which  this  change  in  rigidity  has  upon 
the  distribution  of  bending  moment. 

Take  as  an  example,  a  floor  supported  on  columns  of  equal 
spacing,  say  17  feet  center  to  center,  with  an  8  inch  rough  slab  and 
assume  a  thickness  of  finish  coat  of  1J  inches  added  thereto.  With 
slab  rods  f  inches  diameter  in  four  layers,  the  distance  from  center 
of  the  slab  steel  at  the  support  to  the  bottom  of  the  slab  would  be 
approximately  7  inches,  while  at  mid  span  with  the  rough  slab  the 
distance  from  the  center  of  the  slab  steel  tothe  top  of  the  slab  would 
be  about  7.3  inches,  or  with  the  finished  slab  8J  inches  the  relative 
rigidity  of  the  central  plate  would  be  as  the  squares  respectively 
of  7.3  and  8.5  or  as  49  is  to  72.25,  or  a  ratio  of  I  A7.  Now  the  moment 
will  be  distributed  between  mid  span  and  the  supports  in  such  manner 
that  it  is  divided  in  proportion  to  the  rigidities  of  the  cantilever  and 
suspended  span  and  in  such  manner  that  half  the  sum  of  the  moments 
over  the  supports,  plus  the  moment  at  mid  span  is  constant,  and 
equals  W  L  /S.  Consequently  this  change  in  relative  rigidity 
would  reduce  the  applied  moment  and  the  unit  steel  stresses  at  the 
support  by  twelve  percent,  and  would  increase  the  applied  moment 
to  be  resisted  at  mid  span  by  approximately  23  percent.  But  this 
increase  of  the  applied  moment  at  mid  span  would  increase  the 
unit  stress  in  the  steel  at  mid  span  either  not  at  all  or  very  slightly, 
because  the  increase  of  slab  thickness  due  to  the  finish  will  not  only 
increase  d  but  also  j  so  that  the  arm  jd  of  the  steel  and  its  moment 
of  resistance  will  be  so  increased  as  to  leave  the  unit  stress  in  the 
steel  substantially  the  same. 

10.  Illustrative  Example.     Compute  the  bending  moment  over 
the  column  in  the  case  of  the  Northwestern  Glass  Company  Build- 
ing*, at  Minneapolis: 

Diameter  of  the  basement  columns  30  inches.  Virtual  cap  48 
inches;  extreme  outside  diameter  54  inches.  Slab  reinforcement, 
four  belts  of  fifteen  f  inch  rounds  each  way,  7'  9"  wide.  The  central 
line  of  columns  had  no  lap  of  slab  rod  belts 

The   diameter  between  bearings  is   13'   net.     Net   span  equals 

*Described  in  a  paper  by  H.  T.  Eddy,  to  the  American  Society  of  Civil 
Engineers,  Vol.  LXXVII,  p.  1338,  1914. 


COMPUTATION    OF    INFLUENCE    OF    FINISH    COAT  155 

.77  full  span,  center  to  center  of  columns.  Therefore,  the  applied 
moment  at  the  support 

=  .77  W  L/120  =  TFL/15.6 

Radial  moment  is  half  the  applied  moment,  =  W  L/31.2. 
The  load  was  95,000  pounds. 

M  =  95,000X17X12  =  620,000  inch  pounds. 
31.2 

The  center  of  action  of  steel  was  6.4  inches  from  the  bottom  of  the 
slab,  and  jd  equals  5.5  inches.  The  heavy  radial  rods  are  four 
If  inch  rounds.  Section  of  slab  steel  over  the  cap  is  four  square 
inches.  Total  eight  square  inches.  620,000/8X5.5  =  the  average 
of  14,000  pounds  per  inch  on  the  steel.  The  neutral  plane  is  found 
to  be  2f "  from  the  bottom.  The  center  of  the  slab  steel  is  0.6  inches 
above  the  center  of  action  of  steel  stress,  so  that  the  mean  slab  steel 
stress  is  as 

2.75  :  3.35  or  as  14,000  Ibs.  :  16,000  Ibs. 
and  the  steel  stress  in  the  large  rods  is  to  the  mean  stress  as 

2.75  :  2.  or  as  14,000  Ibs.  :  10,300  Ibs. 

These  computations  are  fair  approximations  within  the  limit  of  error 
involved  in  the  assumptions  made,  namely,  that  the  coaction  of  the 
large  rods  with  the  concrete  is  equivalent  to  uniform  distribution 
of  their  area  about  the  circumference.  This  cannot  be  precisely 
true  because  the  stresses  from  bond  shear  are  distributed  thro 
the  mass  in  proportion  to  the  square  of  their  distance.  This  rela- 
tion indicates  an  interesting  variation  of  stress  about  the  head.  The 
form  of  the  neutral  plane  is  no  longer  a  true  plane  but  is  scalloped 
in  form,  approaching  nearer  the  middle  of  the  slab  under  the  large 
rods  and  being  further  removed  from  the  middle  of  the  slab  between 
the  large  rods.  Investigation  of  the  error  involved  in  the  assump- 
tion that  the  neutral  surface  is  a  true  plane,  however,  will  show  that 
the  error  is  not  large,  making  a  reduction  of  not  more  than  ten  per 
cent  in  the  steel  stress  in  the  large  rods,  making  a  small  increase  in 
the  concrete  compression  and  very  little  change  in  the  stress  in  the 
slab  steel.  The  results  above  obtained  check  closely  with  experi- 
mental determinations. 

Thus  it  is  obvious  that  with  full  lapping  of  every  belt  there  is  a 
large  excess  of  steel  in  the  standard  Mushroom  type  and  this  excess 
of  steel  would  throw  the  line  of  maximum  unit  stress  in  the  slab 
steel  to  another  point  than  that  where  the  greatest  moment  to  be 
supported  occurs. 


156  DEPRESSED    HEADS 

1 1.  Depressed  Head  or  Drop.  The  variation  in  the  moment  at 
the  support  by  reason  of  the  addition  of  strip  fill  and  by  modification 
of  the  diameter  of  the  cap  have  been  considered  and  it  is  next  in  order 
to  investigate  the  modification  in  the  moment  when  the  slab  thick- 
ness is  increased  at  the  column  over  and  above  that  at  mid  span. 

If  the  thickness  of  the  slab  be  increased  by  some  sixty  percent 
as  is  commonly  done  by  a  square  block,  or  drop  head  over  the  capital , 
having  a  size  of  0.4LX0.4L  with  little  or  no  steel  except  the  belt 
rods  without  laps,  then  the  rigidity  is  increased  by  reason  of  the 
increase  of  thickness  in  such  a  way  that  the  lines  of  inflection  are 
thereby  removed  somewhat  from  the  column  centers.  The  slab 
belts  should  not  be  reduced  in  width  below  7  /16  L  =  0.44L  because 
this  width  is  necessary  to  make  the  reinforcement  cover  the  panels 
properly  and  they  should  be  located  at  the  top  of  the  slab  as  far  as 
to  the  extreme  edge  of  the  head,  viz:  as  far  as  to  0.2L  or  0.22L  from 
the  column  center.  This  will  fix  the  lines  of  inflection  at  about  0.27L 
from  column  centers  instead  of  at  0.22L  or  0.23L  as  in  the  Mush- 
room panel.  Such  a  position  of  the  lines  of  inflection  which  en- 
larges the  cantilever  areas  will  decrease  the  moments  at  mid  span 
and  increase  those  at  the  edge  of  the  cap.  The  thickness  of  the 
slab  at  mid  span  may  be  safely  reduced  by  ten  percent  for  this  reason, 
but  the  applied  moments  at  the  edge  of  the  cap  should  not  in  this 
case  be  taken  as  less  than  W  L  /1 2,  instead  of  W  L  /1 5  previously 
used  for  slabs. 

With  designs  of  this  character  little  or  no  increase  in  steel  has 
usually  been  provided  over  and  above  the  slab  rod  steel  and  such 
designs  are  likely  to  show  weakness  over  the  cap.  This  weakness 
has  too  frequently  been  assumed  to  apply  also  to  the  Mushroom 
system  which  is  almost  invariably  over-reinforced  at  this  important 
critical  section. 

The  effect  of  moving  out  the  line  of  inflection  is  evidently  to 
increase  the  bending  moment  brought  upon  the  column  by  unbal- 
anced loads  and  to  decrease  the  toughness  of  the  construction  by 
the  introduction  of  a  sharp  angle  where  the  depressed  head,  as  it 
is  called,  joins  the  body  of  the  slab.  As  just  stated,  these  depressed 
heads  or  thickening  up  of  the  slab,  are  generally  made  about  0.4 
of  the  span  length  in  size  in  each  direction  and  approximately  .6 
of  the  depth  of  the  slab  in  thickness.  They  permit  a  reduction  of 
perhaps  ten  percent  of  the  thickness  in  the  main"  slab,  which  is 
almost  exactly  off-set  by  the  addition  of  concrete  for  the  depression. 


SUMMARY  157 

The  forms  for  this  design  will  cost  more  than  for  the  Mushroom 
design,  while  some  saving  in  steel  can  be  effected  where  the  columns 
are  of  sufficient  size,  this  saving  in  steel  is  more  than  off-set  by 
increased  cost  of  centering  and  reduced  toughness  or  ability  to 
withstand  unbalanced  load  without  serious  over-strain. 

The  developments  in  this  chapter  are  applications  of  the  theory 
of  work  to  bring  out  the  nature  and  characteristics  of  the  predom- 
inant mechanical  actions  in  flat  slabs.  This  method,  while  well 
suited  to  elucidate  the  general  nature  of  these  actions  and  explain 
their  general  properties,  is  lacking  in  precision  in  its  present  form 
and  is  not  so  well  fitted  to  be  the  basis  for  exact  calculation  of  struc- 
tures as  analysis  based  on  the  equilibrium  and  deformation  of  the 
elements  of  the  slab.  The  following  chapters  will  therefore  be 
devoted  to  a  more  rigorous  analysis  of 'the  flat  slab  based  on  the 
conditions  of  equilibrium  and  deformation  of  its  infinitesimal 
elements. 


SSMiiN 

Sflllll   illfrSi*! 


m 


MS 

Jim 


iinii!  mm  is  IBS 

mill 


•*• 


O'REILLY  ESTATE  BUILDING,  ST.  LOUIS,  MO. 

A.  B.  Groves,  Architect  Murch  Bros.  Construction  Co.,  Contractors 

G.  S.  Bergendahl,   M.  AM.,  Soc.  C.  E.,  Engineer,  St.   Louis  Representative,  Mushroom  System 


TISCHERS  CREEK  BRIDGE,  DULUTH,  MINN. 

Spans  are  26  feet  longtitudinally 
This  type  is  built  with  spans  up  to  50'0" 


View  of  Reinforcement  in  Place 

TISCHERS  CREEK  BRIDGE,  DULUTH,  MINN. 
Designed  by  C.  A.  P.  Turner  Geo.  H.  Lounsbury,  Contractor 


CHAPTER  V. 

THEORY  OF  THE  STRENGTH  AND  FLEXURE  OF  THE  STANDARD 
MUSHROOM  TYPE. 

1 .  The  superiority  of  flat  slab  floors  supported  directly  on  columns, 
over  other  forms  of  construction  when  looked  at  from  the  stand- 
point of  lower  cost,  better  lighting,  greater  neatness  of  appearance, 
and  increased  safety  and  rapidity  of  constraction,  is  so  generally,  or 
rather  so  universally  conceded  as  to  render  any  reliable  information 
relative  to  the  scientific  computation  of  stresses  in  this  type  of  con- 
struction of  great  interest.  Heidenreich,  in  his  Engineer's  Pocket 
Book  on  Reinforced  Concrete,  page  89,  classifies  this  type  as  floors 
without  beams  and  girders — " Mushroom  System." 

Since  "mushroom,"  as  applied  to  concrete,  is  an  arbitrary  or 
fanciful  term,  and  indeed,  almost  a  contradictory  one,  a  word  of 
explanation  as  to  its  origin  may  be  of  interest.  The  term  was 
originated  by  C.  A.  P.  Turner,  of  Minneapolis,  and  applied  to  his 
flat  plate  construction,  more  particularly  because  of  the  fancied 
resemblance  to  the  mushroom,  of  the  column  and  column  head 
reinforcement  of  that  particular  form  of  his  flat  plate  construction 
which  he  seemed  to  prefer  by  reason  of  certain  practical  advantages. 
Another  fancied  resemblance  is  the  rapidity  of  erection,  comparable 
to  the  over-night  growth  of  the  mushroom.  Here  the  resemblance 
ceases,  since  the  construction,  once  erected,  is  enduring  and  per- 
manent. 

The  Mushroom  System  is  a  continuous  flat  plate  of  concrete 
supported  directly  on  columns,  and  reinforced  in  such  a  maner  that 
circular  and  radial  tensile  stresses  concentric  with  the  column  are 
provided  for  by  metal  reinforcement  in  the  tension  zone  above  the 
columns,  and  similar  provision  is  made  for  tensile  stresses  in  the 
lower  portion  of  the  slab  concentric  with  the  center  of  the  panel, 
diagonally  between  the  columns.  Since  all  forces  in  a  plane  may  be 
resolved  into  equivalent  components  along  any  pair  of  axes  at  right 
angles  to  each  other,  it  is  possible  to  provide  reinforcement  to  resist 
any  horizontal  tensile  stresses  in  the  slab  by  various  arrangements 
of  intersecting  belts  of  rods  at  zones  where  these  stresses  occur. 


REINFORCEMENT    IN    THE    MUSHROOM    SYSTEM 


159 


All  arrangements  of  this  kind  are    by  no    means    equally  effective. 
A   system   of  wide   reinforcing    belts  from  column  to  column  com- 


Fig.  53.     Vertical  Section  of  Standard  Mushroom  Head  showing  position  of  Radial  and  Ring 

Rods,    and    Slab    Rods,    Vertical   and   Horizontal    Sections  of  Spirally  Hooped 

Column,  with  Plain  Bar  Hoop  Collar  Band,  Vertical  Reinforcing 

Rods  and  Elbow  Rods. 


Fig.  54.     Plan  of  Reinforcement  in  Standard  Mushroom  System.     Radial  and  Ring  Rods,  Collar 
Band  and  Slab  Rods.     Diameter  of  Head  =g=  -&(a+b). 

bined  with  a  system  of  radial  and  ring  rods  to   constitute  a  large, 
substantial  cantilever  mushroom  head  at  the  top  of  each  column 


160  STANDARD    MUSHROOM    SYSTEM 

provides  a  very  effective  and  economical  arrangement  for  controlling 
the  distribution  of  the  stresses  in  the  slab,  and  furnishes  the  resis- 
tance necessary  to  support  these  stresses  by  placing  the  steel  where  it 
is  most  needed.  It  not  only  has  the  same  kind  of  advantage  that  the 
continuous  cantilever  beam  has  over  the  simple  girder  for  long 
spans,  but  combines  with  it  the  kind  of  superiority  that  the  dome  has 
over  the  simple  arch  by  reason  of  circumferential  stresses  called  into 
play,  which  adds  greatly  to  the  carrying  capacity  of  the  slab. 

In  the  standard  mushroom  type,  which  is  quite  fully  discussed 
in  this  paper,  the  heavy  frame  work,  concentric  with  the  column, 
supports  the  slab  reinforcement  at  a  fixed  elevation,  furnishes  a  high 
degree  of  resistance  to  shear,  and  secures  a  high  degree  of  safety 
during  construction.  It  extends  as  a  cantilever  approximately  one 
fourth  of  the  way  to  the  next  column  as  shown  in  Figs.  53  and  54  on 
page  159.  Arranged  upon  the  radial  rods  of  this  frame  rest  two  or 
more  large  hoops  and  upon  these  rest  the  wide  spreading  belts  of  rods 
which  extend  both  directly  and  diagonally  from  column  to  column. 
Over  the  columns  these  belts  lie  near  the  upper  surface  of  the  slab,  but 
they  run  near  the  lower  surface  as  they  approach  points  midway 
between  columns. 

The  cantilever  slab  thus  formed,  not  only  has  the  same  advant- 
ages for  this  form  of  construction  that  the  cantilever  construction 
has  for  long  span  bridges,  but  it  causes  the  slab  to  have  greater 
stiffness  and  gives  it  greater  resistance  to  shear  in  the  neighborhood 
of  the  columns;  it  removes  the  locus  of  zero  bending  moment  to  a 
much  greater  distance  from  the  column  than  would  otherwise  be  the 
case,  thus  dimininishing  the  area  of  that  part  of  the  slab  which  tends 
to  become  concave  on  its  upper  face  and  enlarging  the  convex  area. 

The  cantilever  frame-work  further,  not  only  moves  the  locus  of 
zero  bending  outward  from  the  column,  but  it  also  fixes  the  locus  of 
zero  bending  moment  at  a  known  position  so  that  it  does  not  vary 
with  increase  and  decrease  of  the  load  or  change  of  the  load  from  one 
span  to  an  adjacent  span  as  would  be  the  case  were  the  mass  of 
metal  in  the  frame  and  its  stiffness  largely  reduced.  This  is  ac- 
complished as  follows: 

The  locus  of  zero  bending  moments  is  fixed  by  the  dip  of  the 
reinforceing  rods  as  they  leave  the  upper  surface  of  the  slab  near 
the  edge  of  mushroom  and  pass  below  the  neutral  surface  to  a  level 
near  the  bottom  of  the  slab.  Such  change  of  tensile  resistance  in 
the  slab  necessarily  localizes  at  these  points  the  zero  bending  mo- 
ments. 


BIBLIOGRAPHY  161 

In  addition  to  the  advantages  just  mentioned,  which  are 
of  so  self-evident  a  character  as  to  be  readily  appreciated  even 
by  the  layman,  there  is  another  of  such  an  obscure  and  apparently 
inexplicable  a  nature  that  it  was  for  years  denied  as  incredible  and 
regarded  as  non-existent  by  practical  builders,  and  engineers  as 
well,  unless  they  had  opportunity  to  be  convinced  of  its  reality 
by  experiment.  We  here  refer  to  the  additional  strength  and  stiff- 
ness which  is  imparted  to  a  belt  of  rods  in  a  given  direction  in  a 
slab  by  another  belt  at  right  angles  to  the  first  belt,  or  at  various 
angles  with  it.  This  should  be  designated  as  slab  action  proper 
in  distinction  from  cantilever  action.  It  depends  for  its  amount 
upon  the  value  of  Poisson's  ratio  of  the  lateral  effect  due  to  direct 
elongation  in  the  slab,  and  is  the  basis  of  the  so  called  circum- 
ferential stresses,  which  make  the  strength  and  stiffness  of  such 
reinforced  flat  slabs  much  greater  than  they  are  estimated  to  be 
when  these  are  neglected,  as  they  usually  have  been.  This  mis- 
taken view  has  in  the  past  constituted  the  most  serious  obstacle 
to  the  adoption  of  this  form  of  structure,  and  has  been  the  ground 
of  conscientious  opposition  to  its  introduction  on  the  part  of  con- 
sulting engineers.  It  is  the  object  of  this  investigation  to  remove 
so  far  as  possible  all  reasonable  uncertainty  as  to  the  rational  theory 
of  this  form  of  structure. 

The  following  partial  bibliography  of  this  subject  may  be  useful 
to  those  unfamiliar  with  what  has  been  done  in  this  field. 

Concrete  Steel  Construction,  (305pp) 

By  C.  A.  P.  Turner,  M.  Am.  Soc.  C.  E. 
816  Phoenix  Bldg.,  Minneapolis,  1909. 

Reinforced  Concrete  Construction,  (259pp) 

By  Turneaure  and  Maurer,  University  of  Wisconsin 
Wiley,  N.  Y.,  1907. 

Concrete,  Plain  and  Reinforced,  (483pp) 
By  Taylor  and  Thompson, 
Wiley,  N.  Y.,  1911. 

Trans.  Am.  Soc.  C.  E. 
Vol.  LVI.  June  1906. 

Engineering  News: — 
Oct.  4,  1906,  p.  361. 
Feb.  18,  1909,  p  176. 
Dec.  23,  1909,  p.  694. 

Engineering  Record: — 

March  2S,  1908,  p  374. 
May  2,  1908,  p  575. 
Oct.  10,  1909,  p  411. 
April  3,  1909,  p  408. 
April  10  .1909,  p  492. 


162 


Notation. 

2.     All  lengths  and  areas  are  measured  in  inches,  and  all  weights 
in  pounds. 
A     =   area  of  cross  section  of  steel  reinforcement  per  unit  width  of 

slab,  in  case  it  be  assumed  to  be  replaced  by  a  uniform  sheet 

of  equal  weight. 

AI  =   area  of  cross  section  of  all  the  rods  in  one  side  belt. 
A2  =   area  of  cross  section  of  all  the  rods  in  one  diagonal  belt. 
a     =   one  half  the  longer  side  of  a  panel  from  center  to  center  of 

columns. 

b      =   one  half  the  shorter  side  of  a  panel. 
B     =   the  shortest  distance  along  one  side  of  a  panel  from  the  edge 

of  a  column  cap  to  the  edge  of  the  next  cap. 
Ci  and  C2  are  constants  depending  on  the  relative  lengths  of  the  sides 

of  any  panel,  which  reduce  to  unity  for  any  square  panel. 
DI  =   the  deflection  of  the  middle  of  the  longer  side  of  the  panel 

below  the  edge  of  the  cap. 
D2  =   the  deflection  of  the  center  of  the  panel  below  the  edge  of  the 

cap. 
d     =   the  effective  thickness  of  the  slab  at  any  point,  being  the 

vertical  distance  from  the  center  of  action  of  the  reinforce- 
ment to  the  compressed  surface  of  the  concrete. 
di    =   the  vertical  distance  from  the  center  of  the  rods  in  the  side 

belt  at  mid  span  to  the  top  surface  of  the  concrete. 
d2    =   the  distance  at  the  center  of  the  panel  from  the  center  of  the 

rods  in  the  second  or  upper  diagonal  belt  to  the    top    of    the 

concrete. 
d-4    =   the  distance  at  the  edge  of  the  cap  from  the  center  of  the  third 

belt  of  rods  from  the  top,  to  the  compressed  surface  of  the 

concrete. 

E  or  E*   =   Young's  modulus  for  steel    =   3  x  107. 
Ec   =   Young's  modulus  for  concrete. 
ei     =   elongation  in  steel  parallel  to  long  side  belt. 
e2     =   elongation  in  steel  parallel  to  short  side  belt. 
ei     =   elongation  in  steel  parallel  to  diagonal  belt 


NOTATION  1()3 

F     =   modulus  of  elastic  resistance  to  shearing. 

/s     =  Ee   =   intensity  of  actual  stress  in  steel. 

/c     =   intensity  of  stress  in  concrete. 

g     =   7/16  (a +6)  =  the  diameter  of  the  mushroom  head  and  width 

of  belts. 

h     =  the  total  actual  thickness  of  concrete  slab. 
id    =   vertical  distance  from  center  of  tension  of  steel  to  neutral 

surface  of  slab. 
jd    =   vertical  distance  from  center  of  tension  in  steel  to  center  of 

compression  in  concrete. 
kd   =   vertical  distance  from  neutral  surface  to  compressed  surface 

of  concrete,  hence  i  +  k  =  1. 
K    =   Poisson's  ratio  of  lateral  effect  due  to  longitudinal  resistance 

in  reinforced  concrete  slabs. 

LI    =   2a   =   long  side  of  panel  between  column  centers. 
L2   =   26   =   short  side  of  panel  between  column  centers. 
/     =   distance  from  collar  band  at  top  of  column  to  edge  of  cap. 
mi  =   true  moment  of  resistance  of  the  tensile  stresses  in  steel  parallel 

to  the  long  side  per  unit  of  width  of  slab. 
m2  =   true  moment  of  resistance  of  steel  parallel  to  short  side  per 

unit  of  width. 

n\i  and  m2  =  apparent  moments  per  unit  of  width  of  forces  applied 
parallel  to  the  long  and  short  sides  respectively. 

n  =  the  apparent  moments  per  unit  of  width  of  the  equal 

twisting  couples  parallel  to  either  side. 

Pi  =   intensity  of  the  forces  applied  parallel  to  the  long  side. 

p<2  =   ditto  for  short  side. 

p  =  intensity  of  stress  in  extreme  fiber  of  radial  rods. 

q  =   load  on  slab  in  pounds  per  square  inch. 

RI  and  R2    =   the  radii  of  curvature  of  vertical  sections  of  the  slab 

parallel  to  the  long  and  short  sides  respectively. 
Si  and  s2       =   the  vertical  shearing  stresses  per  unit  of  width  of  slab 

respectively  perpendicular  to  the  long  and  short  sides 

of  the  slab, 
s  =   the  intensity  of  vertical    shearing    stress    in    radial 

rods. 
t  =   either  of  the  equal  horizontal  tangential  or  shearing 

stresses  parallel  to  the  sides  of  the  panel. 


164 


NOTATION.       SQUARE    PANEL 


t 

u  and  v 

V 

x  y  z 


Az 

Zi  and  z-2 

d 

8z 

5x 


=  the  thickness  of  a  radial  rod. 

=   deformations  parallel  to  the  long  and  short  sides  re- 
spectively. 

=  total  vertical  shearing  stress  in  radial  rod. 

=   horizontal  and  vertical  coordinates  parallel  to  sides 
of  panel. 

=   difference  of  two  vertical  coordinates. 
=   deflections  of  radial  rods. 
=   sign  of  partial  differential. 

=   partial  differential  coefficient  of  z  with  respect  to  x. 


Fig.  55.     Plan  of  Reinforcement  Mushroom  System.     Square  Panel,  g-=\L  (as  drawn). 
Line  of  Ultimate  Weakness. 


TRUE    AND    APPARENT    STRESSES  16") 

3.  As  preliminary  to  a  general  investigation  of  the  rational 
analysis  of  the  flat  slab,  it  seems  desirable  in  the  first  place  to 
make  a  brief  exposition  of  the  relationship  between  the  true  bend- 
ing moments  and  the  apparent  bending  moments  in  the  flat  slab  as 
follows : 

The  fundamental  equations  of  extensional  stress  and  strain  in 
thin  flat  plates  and  slabs,  established  a  generation  ago  and  accepted 
by  Grashof*  and  by  all  authorities  on  the  subject  since  then,  maybe 
written  in  the  forms: 

Eel  =  p1—Kp2  \_  (^ 

Ee2  =  p2  —  Kpl  )  ' 

(l-K2)Pl  =  E(el+Ke2)\  (la) 

(1  -  K2)P2  =  E(e2  +  Kej  '/  ' 

in  which  pi  and  p2  are  the  external  applied  or  apparent  stresses  per 
unit  of  area  of  cross  section  of  the  plate,  or  of  the  reinforced  slab, 
which  act  parallel  to  the  axes  of  x  and  y  respectively  if  these  latter 
lie  in  the  neutral  plane  of  the  slab;  and  e\  and  e2  are  extensometer 
elongations  of  plate  or  slab  reinforcement  per  unit  of  length  parallel 
to  x  and  y  respectively.  E  is  Young's  modulus  of  elasticity,  and  K 
is  Poisson's  ratio  of  the  lateral  effect  due  to  linear  elongation.  Any 
piece  of  material  which  is  subjected  to  stress,  and  is  of  such  shape 
that  more  than  one  of  its  dimensions  is  considerable,  as  compared 
with  its  remaining  dimension,  must  have  its  stresses  and  strains 
considered  with  reference  to  the  lateral  effect  produced.  This  is  the 
case  in  plates  and  slabs,  as  it  is  not  in  case  of  rods  and  beams. 

In  the  above  equations  Ee\  and  Ee2  are  the  true  stresses  per  square 
inch  of  section  of  reinforcement  acting  along  lines  parallel  to  x  and  y 
respectively,  whatever  pl  and  p2  may  be.  These  latter  are  the  cause 
of  true  stresses,  but  are  not  themselves  the  values  of  the  true  stresses, 
as  they  are  in  case  of  rods,  etc.,  where  one  dimension  only  is  large. 

These  equations  show  that  the  elongation  e\  in  the  direction  of 
x  is  not  dependent  alone  upon  the  tension  pi  applied  in  that  direc- 
tion, for  it  is  diminished  by  any  tension  acting  along  y,  but  is  in- 
creased by  any  compression  acting  along  y.  It  thus  appears  that 
any  tension  p2  along  y  assists  the  piece  in  resisting  elongation  along 
x  and  makes  it  able  to  endure  safely  a  larger  applied  stress  p\  with 
the  same  degree  of  safety,  i.  e.,  with  the  same  percentages  of  elonga- 
tion or  true  stress.  But  it  is  also  equally  true  that  any  compression 
of  amount  p2  reduces  the  safe  value  of  pi  which  may  be  applied  to 

*Theorie  der  Elasticitat  und  Festgkeit,  F.  Grashof  Berlin  1878. 


166  TRUE    AND    APPARENT   MOMENTS 

it.  These  principles  are  not  in  accordance  with  those  which  hold 
in  ordinary  computations  for  rods  and  bars,  whose  lateral  dimensions 
are  small  compared  with  their  lengths,  and  whose  lateral  stresses 
are  negligible.  This  divergence  between  the  true  stresses  as  shown 
by  actual  deformations,  and  the  apparent  or  applied  stresses,  is  a 
fruitful  source  of  error  in  the  attempted  computation  of  slabs. 

Equations  (1)  in  their  present  form  apply  to  simple  extensional 
or  compressive  stresses  and  strains  but  may  be  extended  to  apply 
to  bending  of  slabs  in  the  following  manner  : 

Take  A  as  the  cross  section  of  the  reinforcement  per  unit  of 
width  of  slab  when  the  actual  reinforcement  is  regarded  as  distrib- 
uted into  a  thin  sheet  of  uniform  thickness,  and  let  jd  be  the  vertical 
distance  from  the  center  of  the  reinforcement  to  the  center  of  com- 
pressional  resistance  of  the  concrete  regarded  as  a  fraction  j  of  d, 
d  being  the  distance  from  the  center  of  the  steel  to  the  top  of  the 
slab.  Then 

ni!  =  Api  jd,  and  m2  =  Ap2  jd,  ...................  (2) 

are  the  apparent  bending  moments  per  unit  of  width  of  slab,  of  the 
applied  apparent  stresses  pi  and  p2,  tending  when  positive,  to  cause 
lines  which  before  bending  are  straight  and  parallel  to  x  and  y  re- 
spectively, to  become  concave  upwards. 

Again  mi  =  Ee±  Ajd,  and  ra2  =  Ee2  Ajd,  .................  (3) 

are  the  true  bending  moments  of  the  actual  resistance  stresses  in 
the  reinforcement  per  unit  of  width  of  slab,  as  shown  by  extenso- 
meter  strains  in  the  steel  parallel  to  the  axes  of  x  and  y  respectively. 
Multiply  equations  (1)  thru  by  Ajd  and  substitute  the  values 
given  in  equations  (2)  and  (3),  from  which  we  obtain  the  following 
relations  between  the  true  and  apparent  bending  moments  in  the  slab. 

mi  =  mi  —  Km2\ 
•  m2  =  m   — 


(1  —  K2)ml  =  ml  +  Km2  \ 
(l—K2)m2  =  m2 


These  equations  bring  out  in  a  striking  manner  the  essential  diver- 
gence of  the  correct  theory  of  slab  action  from  that  of  beam  action 
in  which  latter  case  we  have  the  well  known  equations 

ml  =  mi,  and  m2  =  m2 

i.  e.,  in  beams  the  moment  of  the  applied  forces  is  equal  to  the 
moment  of  the  internal  resistance,  which  is  not  true  in  slabs. 


TRUE    AND    APPARENT   MOMENTS  1()7 

All  attempts  to  base  computations  of  the  deflection  of  slabs 
upon  beam  action  are  therefore  necessarily  erroneous.  Such  com- 
putations are  inapplicable  and  misleading,  hence  deflections  and 
stresses  in  slabs  cannot  be  correctly  computed  by  any  form  of 
simple  or  compound  beam  theory. 

Equations  (4)  show: 

1st  That  at  points  where  nij  and  m2  are  of  the  same  sign,  (as 
for  example  in  the  convex  part  of  the  mushroom  near  the  columns 
and  also  near  the  center  of  the  panel)  the  true  bending  moments 
mi  and  w2,  which  determine  the  actual  stresses  in  the  reinforcement 
are  less  than  the  apparent  bending  moments,  which  latter  have  been 
ordinarily  assumed,  according  to  the  beam  theory,  to  determine 
those  stresses. 

2nd  That  the  compressive  stresses  in  the  concrete  around  the 
column  cap  are  determined  on  the  same  principles  as  the  tensile 
stresses  and  are  consequently  reduced  in  accordance  with  the  value 
of  K  by  a  considerable  percentage  below  values  corresponding  to 
ni!  and  ni2  of  the  beam  theory. 

3rd  That  at  points  where  n^  and  m2  have  different  signs,  as 
they  have  for  example  in  the  middle  part  of  the  span  at  the  side  of 
the  panel  directly  between  mushroom  heads,  the  values  of  the  true 
bending  moments  are  larger  than  the  apparent  moments  as  found 
by  the  beam  theory. 

4th  One  deduction  from  this  (which  is  also  confirmed  by 
extensometer  tests)  is,  that  in  slabs  having  equal  side  and  diagonal 
belts  of  reinforcing  rods  the  greatest  actual  extensions  and  true 
stresses  in  the  steel  occur  at  the  mid  points  of  those  reinforcing  rods 
which  run  directly  between  the  mushroom  heads  parallel  to  the 
sides  of  the  panel,  and  do  not  occur  at  the  center  of  the  panel  where 
ni!  and  m2  have  their  greatest  values.  Further,  the  true  stresses  in 
the  reinforcement  are  not  so  large  at  the  edge  of  the  column  caps  as 
at  the  points  just  indicated.  Neither  of  these  conclusions  is  in 
accordance  with  the  beam  theory  as  implied  in  ordinary  formulas 
such  as  have  been  frequently  adopted  in  practice  in  computing  slabs. 

5th  In  making  any  statement  or  specification  respecting  the 
bending  moments  at  any  point  of  a  slab,  it  is  essential  to  state  which 
bending  moments  are  contemplated,  the  true  bending  moments  or 
the  apparent  moments,  with  the  understanding  that  the  true  bend- 
ing moments  only  are  to  be  used  in  determining  cross  sections  and 
stresses  of  steel.  Any  statement  omitting  this  distinction  is  ambig- 
uous, and  any  requirement  seeking  to  proportion  cross  sections  of 
steel  to  apparent  stresses  and  apparent  moments  is  incorrect. 


168  POISSON'S  RATIO 

4.  Poisson's  ratio  K  plays  an  important  role  in  the  theory  of 
flat  slabs  and  plates,  as  is  evident  from  equations  (1)  and  (4).  Few 
attempts  have  been  made  to  determine  K  by  directly  measuring  the 
amount  of  the  lateral  effect  accompanying  the  elongation  of 
test  specimens,  and,  were  such  measurements  made,  the  relative 
dimensions  of  the  cross  section  of  the  specimen  would  need  to  be 
considered  as  affecting  in  a  very  complicated  way  the  true  value  of 
K  to  be  derived  from  observation.  Reliable  determinations  of  K 
usually  depend  upon  observations  of  Young's  modulus  of  elasticity 
E  and  the  shearing  modulus  of  elasticity  F. 

It  is  proven  in  the  general  theory  of  the  deformation  of  isotropic 
elastic  solids  that  all  the  elastic  properties  of  any  such  solid  are 
determined  without  excess  or  defect  by  its  values  of  E  and  F,  and 
that  Poisson's  ratio  is  a  function  of  E  and  F  expressed  by  the  equation 

K  +  1  =  \E/F (5) 

There  is  evidence  to  show  that  for  concrete  K  is  approximately 
0.1*.  For  steel  it  is  known  that  K  =  0.3  nearly. 

Now  it  is  evident  that  a  horizontal  slab  of  reinforced  concrete, 
in  which  the  reinforcement  consists  of  rods,  differs  from  one  in  which 
its  reinforcement  is  considered  to  be  a  simple  uniform  sheet  of  metal 
in  this,  that  the  former  has  much  less  shearing  rigidity  in  resisting 
horizontal  forces  than  the  latter,  for  in  it  all  stresses  transmitted 
from  one  band  or  belt  of  rods  to  any  other  belt  crossing  it  are  trans- 
mitted thru  concrete  only,  as  is  not  the  case  if  the  reinforcement 
consists  of  a  continuous  sheet.  It  is  evident  therefore  that  the  value 
of  K  which  must  be  employed  in  applying  the  foregoing  equations  to 
reinforced  concrete  slabs  must  exceed  0.3,  the  value  required  in  case 
the  reinforcement  is  a  sheet  of  steel. 

This  analysis  of  the  conditions  affecting  the  value  of  K  for  a 
reinforced  flat  slab  differs  radically  from  assuming  at  random  that 
because  K  =  0.3  for  steel  alone  and  K  =  0.1  possibly,  for  concrete 
alone,  that  therefore  some  intermediate  value  of  K  may  be  correct 
for  these  two  materials  combined  in  a  slab.  Such  an  assumption 
is  merely  a  blind  guess  and  has  no  rational  basis. 

As  already  partly  stated,  the  view  here  put  forth  is  this:  Since 
in  any  homogeneous,  isotropic,  elastic  material  the  experimental 
values  of  E  and  F  perfectly  define  all  its  elastic  properties,  and  since 
we  are  evidently  at  liberty  to  assume  our  flat  slab  as  sufficiently  fine 
grained  in  its  structure  to  act  nearly  like  a  slab  constructed  of  some 
sort  of  homogeneous  materials,  it  will  be  possible  to  determine 

*  Turaeaure  and  Maurer's  Reinforced  Concrete  Construction  2nd  Ed.  1907,  p.  210. 


POLSSON'H  RATIO  169 

certain  mean  values  of  E  and  F  which  will  define  its  elastic  proper- 
ties. It  is  moreover  evident  that  in  a  slab,  where  two  kinds  of  elastic 
solids  are  combined  as  they  are  here,  the  mean  value  of  F  for  the 
combination  is  much  more  affected  by  the  concrete  than  is  E,  which 
latter  may  be  taken  as  that  applying  to  the  steel  alone,  and,  conse- 
quently as  unchanged  by  the  combination.  It  is  otherwise,  however, 
with  F,  because  the  arrangement  of  the  combination  is  such  as  to 
require  the  assumption  of  a  value  of  F  lying  somewhere  between 
that  for  steel  and  that  for  concrete.  Since  the  latter  value  is  much 
less  than  the  former,  the  mean  value  of  F  is  smaller  than  for  steel 
alone. 

This  reasoning  and  other  independent  theoretical  and  kinemat- 
ical  considerations  have  led  to  the  same  conclusion,  viz:  that  the 
correct  value  of  K  for  the  slab  is  larger -than  0.3. 

Assuming  E  =  30,000,000,  we  may  compute  corresponding 
values  of  K  and  F  from  (5)  as  follows : — 

If  K  =  0.1  ,  F  =  13,600,000 
If  K  =  0.3  ,  F  =  11,600,000 
If  K  =  0.5  ,  F  =  10,000,000 

Were  a  perfectly  complete  and  accurate  mathematical  theory 
of  the  flat  slab  at  our  disposal,  we  might  consider  every  experimental 
test  of  the  deflection  of  such  a  slab,  and  every  extensometer  measure- 
ment of  its  reinforcing  rods  as  an  experiment  for  determining  the 
numerical  value  of  K,  since  deflections  and  extensions  would  then 
all  be  known  functions  of  K.  Having  brought  such  a  rational 
theory  to  a  somewhat  satisfactory  degree  of  perfection,  we  have 
found  that,  in  the  light  of  all  known  tests  of  slabs,  the  value  that  best 

satisfies  all  conditions  is      K  =  0.5 (6) 

It  is  possible  that  this  value  of  the  constant  K  for  slabs  may  need 
some  slight  modifications  hereafter,  but  for  the  present  this  may  be 
regarded  as  substantially  correct  for  mushroom  slabs.  It  may  be 
found  necessary  to  assume  a  somewhat  different  value  for  other  forms 
of  structure,  as  for  example,  beam  and  girder  construction.  That, 
however,  must  be  determined  later.  Moreover  it  must  be  said  that 
this  value  of  K  applies  to  tests  made  upon  slabs  from  2  to  4  months 
old,  and  under  loads  which  have  been  applied  to  such  relatively  soft 
concrete  as  this  for  a  period  of  usually  not  longer  than  one  or  two 
days,  and  of  an  intensity  such  as  to  cause  a  maximum  stress  in  the 
steel  of  from  10,000  to  16,000  Ibs.  per  square  inch.  Less  loads  on 
better  cured  concrete,  or  longer  time  under  load,  may  show  con- 
siderable deviation  from  this  value  of  K. 


170  EQUILIBRIUM    OF    SLAB    ELEMENT 

How  important  a  factor  K  is  in  slab  theory  is  evident  on  con- 
sidering equations  (4)  which  show  that  in  a  square  panel  uniformly 
loaded  the  true  moments  as  shown  by  the  elongations  of  the  rein- 
forcing rods  at  the  center  of  the  panel  and  over  the  centers  of  the 
columns  are  only  one  half  the  corresponding  apparent  moments 
derived  from  considering  the  moments  required  to  hold  the  applied 
forces  in  equilibrium,  this  being  on  the  assumption  of  course  that 
K  =  0.5.  (See  further  remarks  in  Section  1,  Chapter  VII). 

5.  In  order  to  derive  the  general  differential  equation  of  shears 
and  moments  in  any  rectangular  panel  in  an  extended  horizontal 
plate  or  slab,  take  the  axes  of  x  and  y  in  the  neutral  plane  of  the 
plate  and  parallel  respectively  to  the  longer  and  shorter  sides  of  the 
panel  with  the  origin  at  its  center  before  flexure  occurs,  and  assume 
that  they  remain  fixed  with  reference  to  the  points  of  support  of  the 
panel.  Then  during  flexure  the  center  of  the  panel  and  all  other 
points  of  the  slab  or  plate  not  in  contact  with  the  fixed  points  of 
support  will  attain  some  deflection  z,  of  amount  to  be  determined 
later.  Take  z  positive  downwards. 

Then  dxdy  is  the  horizontal  area  of  an  element  of  the  slab 
bounded  by  vertical  planes,  and  if  d  be  the  effective  thickness  of  the 
slab  or  plate,  the  areas  of  the  sides  of  this  element  which  are  respec- 
tively perpendicular  to  x  and  y  are  d8y  and  ddx,  while  ddxdy  is  the 
volume  of  the  element. 

We  proceed  to  obtain  the  equations  of  equilibrium  of  this  ele- 
ment of  the  slab  as  follows : — 

Let  Si  and  s2  be  the  total  vertical  shearing  stresses  per  unit  of 
width  of  slab  for  sections  perpendicular  to  x  and  y  respectively.  In 
case  these  shears  are  variable,  as  they  are  in  a  continuously  loaded 
slab,  they  respectively  contribute  elementary  forces  tending  to  move 
the  element  vertically,  of  the  following  amounts : 

d  Si  6  s2 

dydx,         and  -  dxdy 


d  x  d  y 

Assume  that  the  slab  carries  a  uniformly  distributed  load  of  q  pounds 
per  square  unit  of  area.  Then  the  load  upon  the  elementary  area 
5x8y  is  q8xdy,  and  the  equation  of  equilibrium  of  the  vertical  forces 
acting  on  the  element  reduces  to  this: 

d  si  5  s2 

+  +9  =  0 (7) 

0    X  0V 


EQUILIBRIUM    OF    SLAB    ELEMENT  171 

in  which  Si  and  s2  are  taken  as  positive  when  they  are  such  as  would 
be  produced  in  the  slab  by  the  loading  q  in  case  it  were  supported  at 
the  origin  only. 

Let  nij  and  m2  be  the  apparent  moments  per  unit  of  width  of 
slab  of  the  applied  forces  which  tend  to  bend  those  lines  in  the  slab 
which  before  bending  are  parallel  to  x  and  y  respectively.  Take 
them  as  positive  when  they  tend  to  make  those  lines  respectively 
concave  upwards.  These  are  the  moments  obtained  by  multiplying 
the  total  applied  tension  per  unit  of  width  of  slab  by  the  vertical 
distance  jd  from  the  center  of  the  reinforcement  of  the  slab  to  the 
center  of  compression  in  the  concrete  as  given  in  (2) .  These  moments 
are  not  identical  in  a  slab  with  the  true  resisting  moments  m-^  and  ra2 
in  the  same  directions,  which  latter  are  the  moments  obtained  by 
multiplying  jd  by  the  actual  tension  in  the  steel  per  unit  of  width  of 
slab,  which  last  is  to  be  correctly  computed  by  taking  the  product 
of  the  area  of  steel  per  unit  of  width  and  its  elongation  multiplied 
by  E  its  modulus  of  elasticity  as  shown  in  (3). 

Again,  let  n  be  the  twisting  moment  per  unit  of  width  of  ver- 
tical section  of  slab  cut  by  planes  perpendicular  to  either  x  or  y,  and 
acting  about  either  x  or  y,  which  moment  n  is  regarded  as  due  to  the 
variation  of  the  vertical  shearing  stress  sx  when  y  varies,  and  to  the 
variation  of  s2  when  x  varies.  The  moment  n  is  held  in  equilibrium 
by  horizontal  shearing  stresses  in  these  same  sections,  which  are 
opposite  in  sign  above  and  below  the  neutral  surface.  Let  t  be  the 
total  horizontal  shearing  stress  per  unit  of  width  of  slab  in  the  rein- 
forcement on  one  side  of  the  neutral  plane,  then: 

n  =  t  A  j  d (8) 

At  any  point  xy  this  horizontal  shearing  stress  t  must  be  the  same 
for  the  section  perpendicular  to  x,  as  for  the  section  perpendicular 
to  y,  because  in  every  state  of  stress  the  tangential  components  are 
equal  and  of  opposite  sign  on  any  two  planes  mutually  at  right 
angles.  Consequently  the  moment  n  is  the  same  about  x  as  about 
y,  as  has  been  assumed  in  (8). 

It  is  implicitly  assumed  in  (2)  and  (3)  that  the  concrete  on  the 
same  side  of  the  neutral  plane  as  the  reinforcement  is  ineffective 
and  that  its  resistance  is  negligible,  so  that  on  that  side  the  resistance 
of  the  reinforcement  alone  counts.  This  condition  actually  occurs 
only  after  a  state  of  quite  considerable  stress  obtains,  and  of  itself 
affords  a  sufficient  reason  why  the  formulas  based  on  it  fail  of  accu- 
rately representing  deflections  and  elongations  at  small  loads  and 
low  stress. 


172  DIFFERENTIAL   EQUATION    OF    MOMENTS 

The  elementary  couples  acting  on  the  vertical  faces  of  the 
element  which  are  in  equilibrium  with  those  arising  from  the  shear- 
ing stresses  are : — 

(5  nix  6  n    \ 

-f-  -  I     8x3y     about  y,  and 

dx  d  y    / 

(8  m2  5  n   \ 

•    +  -  I     dxdy     about  x; 

d  y  d  x    / 

while  those  arising  from  the  shears  themselves  are : — 
$i  dx  by     and     s2  dx  by. 

Consequently  the  equations  of  equilibrium  of  the  couples  acting  on 
the  element  reduce  to  the  following: 


b  nil  b  n 

r  +  ~r 

o  x  o  y 

8  m2  d  n 


x 


(9) 


Differentiate  equations  (9)  with  respect  to  x  and  y  respectively 
and  substitute  in  (7),  and  we  obtain 

52ni!  52  n  d2  m2 

+  -   q (10) 


d  x2  dx  by  b  y 

which  is  a  general  differential  equation  of  the  apparent  moments 
of  the  applied  forces  which  exist  in  a  uniformly  loaded  slab  in 
terms  of  rectangular  coordinates.  From  it  the  differential  equa- 
tion of  the  deflections  may  be  derived  as  follows: — 

6.  To  obtain  the  general  differential  equation  of  the  deflec- 
tions of  a  slab,  note  that  from  geometrical  considerations  such  as 
are  familiar  in  the  theory  of  beams  we  have 

-RI€I  =  id  =  R2e2..... ?...(!!) 

in  which  R1  and  R2  are  the  radii  of  curvature  of  sections  of  the 
neutral  surface  by  vertical  planes  parallel  to  x  and  y  respectively; 
and  id  is  the  distance  from  the  center  of  the  reinforcement  to  the 


MOMENTS    AND    CURVATURES    IN    SLAB 


173 


neutral  surface.     In  equations   (la)  replace  pi  and  p2  by  values 
given  in  (2),  and  ei  and  e2  by  values  taken  from  (11)  and  we  have: — 


(1  —  Kz)m,  =  E  Aijd*  I-  -  +  - 

\  RI      R2 

(12) 

/I          K  > 
(1  —  #2)m2  =  #  A  yd2  |  -  -  +  - 

But  from  the  theory  of  curvature 

1  d2  z  1  d2  z 

— ,  and  -  —    (13) 

7~>  _Z  7">  rt  Z 

/LI                 o  x                  ri2                0  y 
Also  write  for  brevity         I  =  A  i  j  d2 (14) 

Then  we  have  from  (12),  (13)  and  (14): 

,          /  d2  z  d2  z    \  ' 

(l-#2)mi=±tf/r    —     +K-  —  \ 
\  d  x2  5  y2  / 

.  .  .  (15) 

/  d2  z  52  z   \ 

(1  -  ^2)m2  =  ±  El  I  +K-  ] 

\  &  y2  d  x2   / 

By  the  fundamental  equations  of  elasticity  we  also  have 

(5u       dv\ 
—  +  —  ) (16) 
by      dx/ 

in  which  F  is  the  shearing  modulus,  e3  is  the  horizontal  shearing 
deformation  of  the  reinforcement  for  two  vertical  planes  one  unit 
apart  horizontally,  and 

_l_        dz  _i_         dz 

u  =  ±  id  -      ,    v  =  ±  id  — (17) 

dx  dy 

are  the  deformations  along  x  and  y  respectively,  due  to  the  vertical 
distance  i  d  of  the  reinforcement  from  the  neutral  surface. 

From  (16)  by  help  of  (17)  we  have 

d2  z 

t  =  ±  2  F  id  - (18) 

dxdy 


174  DIFFERENTIAL   EQUATION    OF    DEFLECTIONS 

In  (18)  replace  F  by  its  value  obtained  from  (5),  and  then  sub- 
stitute the  resulting  value  of  t  in  (8)  :  — 
we  then  have 

El         d2  z 
n  =  .....  .......................  (19) 

1  +  K      dxdy 

From  (15)  and  (19)  obtain  values  of  the  second  differential 
coefficients  of  the  moments  appearing  in  (10),  which  on  being  intro- 
duced into  (10),  transform  that  equation  into  the  required  general 
differential  equation  of  deflections  as  follows:  — 

S4  z  5*  z  5*  z          (1  —  K2) 


which  is  a  partial  differential  equation  of  the  fourth  order  that  must 
be  satisfied  by  the  coordinates  x  y  z  of  the  neutral  surface  of  any 
uniform  plate  or  slab  initially  flat,  when  deflected  by  the  applica- 
tion of  a  uniform^  distributed  load  of  intensity  q,  and  supported  in 
any  manner  whatever. 

It  may  be  shown  that  any  deviations  from  strict  accuracy  by 
reason  of  local  stretching  of  the  neutral  surface  (here  neglected)  are 
small  compared  with  corresponding  deviations  in  beam  theory. 

7.  The  solution  of  the  general  differential  equation  of  deflec- 
tions (20)  for  the  case  of  a  horizontal  slab  carrying  a  uniformly 
distributed  load  and  supported  on  rows  of  columns  placed  in  rec- 
tangular array  and  having  the  points  of  support  all  on  the  same 
level,  will  now  be  considered. 

The  integration  or  solution  of  (20)  would,  since  it  is  a  partial 
differential  equation,  introduce  arbitrary  functions  of  the  independent 
variables  x  and  y  whose  forms  would  need  to  be  so  determined  as  to 
cause  the  solution  to  satisfy  the  conditions  imposed  by  the  position 
and  character  of  the  supports  at  certain  points,  or  along  certain 
lines.  It  would  be  possible  to  expand  these  functions  in  terms  of 
ascending  whole  powers  and  products  of  x  and  y,  and,  in  case  the 
supports  are  symmetrically  situated  with  respect  to  the  axes,  the 
expansions  will  contain  no  odd  powers  of  x  or  y,  because  the  value 
of  z  must  remain  unchanged  by  changes  of  sign  of  either  x  or  y,  or 
both  x  and  y.  Any  form  of  polynomial  expansion  which  satisfies 
(20),  and  also  all  the  conditions  of  any  given  case,  must  be  the  correct 
solution  for  that  case,  for,  the  solution  of  any  given  case  must  be 
unique. 


GENERAL   EQUATION    OF    DEFLECTIONS  175 

Instead  therefore  of  carrying  thru  the  tedious  analytical  devel- 
opment involved  in  solving  (20)  mathematically  and  then  applying 
it  to  the  case  we  are  treating,  we  shall  at  once  write  down  the  form 
of  solution  that  applies  to  the  case  in  hand  and  verify  the  fact  that 
it  satisfies  (2)  and  all  the  required  geometrical  conditions.  It  will 
therefore  be  the  solution  sought  for,  which  might  also  have  been 
obtained  by  the  somewhat  intricate  analytical  processes  involved 
in  the  intregation  of  such  differential  equations  as  (20). 

Assuming  at  first  that  the  slab  is  unlimited  in  extent  and  uni- 
form thruout  in  the  distribution  of  its  reinforcement  and  loading, 
and  that  the  parallel  rows  of  supporting  columns  divide  the  slab 
into  equal  rectangular  panels,  we  shall  find  a  solution  in  which  every 
panel  is  deformed  precisely  in  the  same  manner  as  every  other. 
Modifications  made  later  will  render  it  possible  to  take  account  of 
variations  and  irregularities  in  the  distribution  and  arrangement  of 
the  reinforcement,  and  to  estimate  to  some  extent  at  least  the  effect 
of  loading  only  one  or  more  panels. 

Let  2a  be  the  length  and  26  be  the  breadth  of  a  panel;  then  the 
equation  of  its  neutral  surface,  referred  to  axes  parallel  to  its  sides 
and  to  an  origin  fixed  in  space  at  the  center  of  the  neutral  surface  of 
the  panel  before  deflection,  is:  — 

48  EIz  =  q(l  —  K2)  [(a2  -  -  x2)2  +  (62  -  -  y2)2]  .....  (21) 

This  is  the  correct  solution  of  (20)  not  only  because  it  satisfies 
(20),  as  it  will  be  found  to  do  by  trial,  (and  just  as  many  other  func- 
tions of  x  and  y  do  also)  but  it  also  satisfies  all  the  other  conditions 
required  by  the  case  proposed,  viz.: 

1st     z  =  0     when  both  x  =  jli  a  and  y  =  ~^~  b: 
because  there  must  be  no  deflection  at  these  points  of  support  which 
are  on  the  same  level  as  the  origin. 

2nd  dz  /  dx  =  0,  when  x  =  0,  and  also  when  x  =  +.  a;  as  well  as 
dz/dy  =  Q,  when  y  =  0,  and  also  when  y  =  +.6;  because  straight 
lines  drawn  in  space  to  touch  the  slab  across  its  edges,  and  across 
its  mid  sections  parallel  to  those  edges,  must  all  be  horizontal  by 
reason  of  the  symmetry  of  the  slab  on  each  side  of  its  edges  and  mid 
sections.  That  these  conditions  hold  is  evident  from  the  following 
equations  derived  from  (20)  : 


Sx       12  E  I 


,2  2, 

q  x  (x   --a) 


.  .  .  (22) 


176 


TRUE    STRESSES    AND    MOMENTS.       LINES    OF    CONTRA-FLEXURE 


It  is  of  interest  to  note  that  the  sections  of  this  surface  made 
by  all  vertical  planes  parallel  to  the  axes  of  y,  i.  e.,  by  x  =  constant, 
are  precisely  the  same  except  in  position,  since  their  equations  differ 
by  a  constant  only.  The  same  is  true  of  sections  parallel  to  x.  It 
thus  appears,  that,  in  a  square  panel  where  a  =  6,  the  surface  may 
be  regarded  as  a  ruled  surface  described  by  using  the  two  of  these 
curves  on  a  pair  of  parallel  sides  of  the  panel  as  directrices  and  a 
third  one  of  these  curves  as  a  ruler  sliding  on  the  first  two  in  such  a 
manner  as  to  remain  parallel  to  the  other  pair  of  parallel  sides. 

The  deflections  at  the  center  of  the  panel  and  middles  of  the 

sides  are: 

At  x  =  0  =  y,  48  E  I  z  =  q  (1-K2)  (a4  +  64) 

At  x  =  ±  a,  y  =  0,         48  E  I  z  =  q  (1-K2)  b4 

MX  =  0,y  =  ±_b,  48  E  I  z  =  q  (1-K2)  a4 

so  that  in  a  square  panel  the  center  deflection  is  twice  the  mid  edge 

deflection. 

Differentiating  equations  (22)  we  have  by  help  of  (11),  (13), 
(14)  and  (3): 

id  d2  z  (1-K2) 

ei  =  --  =  ±id~  =  -  ~q(3x2-a2) 


id 


5x2 
d2  z 


e-2  =  --  =  +  i  d  - 
R2  dx2 


12  E  A  j  d 

d-K2) 
12  E  A  jd 


^....(23) 


12 


12 


-b2) 


(23a) 


in  which  the  ambiguous  signs  are  to  be  so  taken  that  mi  and  m2  in 
(15)  will  be  positive  at  x  =  0  =  y,  and  negative  at  x  =  it  a  and 

y  =  ±6. 

From  (23)  it  appears  that  extensions  vanish  and  contra-flexure 
occurs  at  lines  lying  in  vertical  planes  whose  equations  are 


and     y  =  ± 


(24) 


It  thus  appears  that  the  slab  is  subdivided  by  these  lines  (24) 
drawn  parallel  to  the  edges  into  a  pattern  which  consists  of  a  rect- 
angle occupying  the  middle  part  of  each  panel,  of  a  size  f  aV3  by 


APPARENT    MOMENTS    AND    SHEARS  177 

f  6^3,  i.  e.,  of  the  same  relative  dimensions  as  the  panel  itself,  and 
bounded  by  lines  (24),  which  rectangle  is  concave  upward  thruout. 

On  all  four  sides  of  this  central  rectangle  are  rectangles  of  saddle 
shaped  curvature  directly  between  the  central  rectangles  of  adjoin- 
ing panels,  while  each  point  of  support  is  situated  in  a  rectangle 
which  is  convex  upward  over  its  entire  area,  of  dimensions 
2a(l—  J  1^3)  by  26(1  —  J 


From  (22)  we  obtain  the  equation 

d2  z  /  dxdy  =  0  ...............................  (25), 

hence  by  (18)  and  (19)  it  follows  that 

t  =  0  =  n,  ....................................  (26), 

from  which  it  appears  that  there  is  no  horizontal  shear  in  the  steel, 
and  no  twisting  moment  in  vertical  planes  perpendicular  to  x  or  y. 
This  would  be  otherwise  evident  from  considerations  of  symmetry. 
It  will  be  shown  that  this  is  not  true  of  all  other  vertical  planes. 

Again  from  (15)  and  (23)  we  have 

2-62)] 
-b2]- 

in  which  we  have  omitted  the  sign    l  as  superfluous. 
From  (9)  by  help  of  (26)  and  (27),  we  have 


—  BI  =  --  =  i  q  x,     and  —  s2  =-   -  =  i  q  y  ......  (28) 

ox  by 

from  which  it  appears  that  any  strip  of  the  panel  parallel  to  x  or  y, 
and  one  unit  wide  exerts  a  shear  at  its  ends  such  as  it  would  if  it  were 
an  isolated  beam  loaded  uniformly  with  an  intensity  of  %q  per  unit 
of  length.  According  to  this,  a  total  shear  of  q  a  b,  which  is  one 
fourth  of  the  total  load  carried  by  the  panel,  appears  at  each  edge  of 
the  panel,  this  total  shear  on  each  edge  being  uniformly  distributed 
along  it. 

It  is  seen  therefore  that  the  form  of  solution  which  we  are 
investigating  implicitly  assumes  that  at  each  edge  of  the  panel  there 
is  some  auxiliary  form  of  structure  that  will  bear  the  shears  coming 
to  it  from  each  side  and  at  the  same  time  assume  the  curvatures  and 
deflections  contemplated  in  (21).  This  will  immediately  engage 
our  further  attention. 


178  SIDE    BELTS 

8.  In  order  to  investigate  more  fully  the  deflections,  stresses 
and  strains  in  the  side  belts  of  any  panel  directly  between  the  mush- 
room heads,  let  us  consider  the  results  just  reached  somewhat  more 
fully. 

The  conclusion  drawn  from  (28)  was,  that  a  panel  with  rein- 
forcement distributed  with  perfect  uniformity  thruout  would  require 
to  be  supported  by  a  narrow  auxiliary  girder  extending  from  column 
to  column  along  each  side,  and  of  such  resisting  moment  as  to  take 
on,  under  its  load,  the  precise  curvature  required  by  the  neutral 
surface  in  (21),  which  curvature  must  be  produced  by  a  uniformly 
distributed  load  of  2  q  a  b,  one  half  of  it  coming  from  each  of  the  two 
panels  beside  it. 

It  seems  then,  that  up  to  this  point,  we  have  in  reality  been 
treating  the  theory  of  the  continuous  uniform  slab  with  specially 
designed  continuous  beams  supporting  its  edges,  without  as  yet 
investigating  those  beams  in  detail.  But  since  no  such  beams  in 
fact  exist  under  the  flat  slab,  it  is  clear  that  the  side  belts  of  the  slab 
lying  directly  between  the  extended  heads  of  the  columns  must 
discharge  the  functions  which  would  be  discharged  by  the  auxiliary 
beams  just  spoken  of.  Such  functions  must  necessarily  be  added 
to  those  already  discharged  by  those  belts  in  supporting  the  loading 
which  rests  directly  upon  them.  In  order  that  this  may  occur  in  a 
manner  readily  amenable  to  analysis,  the  extended  stiffened  head- 
ings of  the  columns  which  constitute  the  mushrooms  should  in 
general  be  approximately  of  the  diameter  required  to  support  the 
ends  of  a  belt  of  reinforcing  rods  forming  a  flat  beam  which  fills  the 
width  along  the  edge  of  two  adjacent  panels  between  the  two  lines 
of  contra-flexure  on  each  side  of  that  edge,  as  given  in  (24). 

This  requires  that  the  mushroom  head  should  have  a  width  of 
at  least  (1  —  J  V^3)  =  .423  of  the  width  of  the  slab  between  col- 
umns. For  reasons  that  will  appear  later,  it  is  current  practice  to 
make  these  heads  not  less  than  •&  =  .437  of  this  width. 

The  lines  of  contra-flexure  in  (24)  have  a  fixity  of  position,  (in  a 
flat  slab  constructed  with  mushroom  heads  of  this  size  and  stiffness,) 
under  single  panel  loads,  that  does  not  exist  in  a  uniform  slab,  or 
where  the  headings  are  not  so  stiff.  It  may  be  readily  shown  by 
Mohr's  theorem  respecting  deflection  curves  as  second  moment 
polygons,  that  where  there  are  large  sudden  changes  in  the  magni- 
tude of  the  moment  of  inertia  7,  such  as  exist  in  this  case  at  the  lines 
of  contra-flexure  at  the  edges  of  the  mushroom,  the  lines  of  contra- 
flexure  remain  fixed.  But  in  systems  where  the  diameter  of  the  head 
is  smaller  than  given  above,  or  its  stiffness  is  much  reduced,  these 


SIDE    BELTS  179 

lines  may  be  removed  to  greater  distances  from  the  center  in  loaded 
panels  surrounded  by  those  not  loaded  than  when  all  are  loaded, 
thereby  increasing  the  deflections  and  stresses  in  a  single  loaded 
panel  over  that  of  a  uniformly  loaded  slab  of  many  panels. 

The  lines  of  contra-flexure  in  (24)  separate  the  slab  into  areas 
which  are  largely  independent  of  each  other,  since  no  bending 
moments  are  propagated  from  one  to  another.  The  only  forces 
crossing  these  lines  of  section  are  the  total  vertical  and  horizontal 
shearing  stresses.  The  horizontal  shears  (which  are  unimportant  so 
far  as  deflections  go)  will  be  considered  later  so  far  as  may  be  necessary, 
but  the  vertical  shears  found  by  (28)  are  of  prime  importance.  Let 
us  then  consider  one  of  these  side  belts. 

In  any  extended  slab  with  its  panels  all  loaded  uniformly 
thruout,  the  vertical  shear  must  vanish  at  all  points  along  sections 
made  by  vertical  planes  thru  the  centers  of  columns  at  each  side  of 
any  panel,  as  appears  by  reason  of  symmetry  of  loads.  Let  the 
edges  of  the  side  belts  be  situated  at  some  given  distances,  say  Xi  and 
yi  on  each  side  of  the  centers  of  all  the  panels,  where  Xi  and  y\  are 
not  necessarily  the  values  of  x  and  y  given  in  (24),  altho  those 
values  are  also  included  in  this  supposition.  Then  by  (28)  there  is 
a  uniformly  distributed  vertical  shear  of  intensity  \  q  y\  along  the 
edge  of  the  belt  at  y  =  yi}  even  tho  the  reinforcement  in  the-  side 
belt  may  be  greater  than  that  in  the  central  rectangle,  for  the  devia- 
tions caused  by  the  irregularity  of  its  distribution  may  be  regarded 
as  unimportant  and  practicably  negligible. 

It  may  then  be  assumed  that  any  side  belt  parallel  to  x  must 
carry,  in  addition  to  that  already  provided  for  in  (21),  a  total  loading 
of  q  yi  per  unit  of  length,  uniformly  distributed  along  the  two  edges 
that  are  parallel  to  x.  Now  since  the  width  of  this  belt  is  2(b  —  yi), 
the  load  already  provided  for  in  (21)  is  \q  per  unit  of  area,  or  q(b-y\) 
per  unit  of  length  parallel  to  x,  which  added  to  that  arising  from  the 
shears  just  mentioned  makes  a  sum  total  of  q  b  per  unit  of  length  of 
belt,  which  it  will  be  noticed  is  independent  of  the  width  of  the  belt. 
In  other  words,  any  such  belt  must  support  a  load  of  one  fourth  of 
the  total  load  on  the  two  panels  of  which  it  forms  a  part,  or  one  half 
of  all  that  lies  between  the  panel  center  lines  which  are  parallel  to 
it  on  either  side.  The  other  half  may  be  regarded  as  carried  to  the 
heads  by  the  diagonal  belbs.  This  in  effect  transfers  the  entire 
loading  of  the  slab  to  the  side  belts  by  the  agency  of  the  shearing 
stresses.  It  does  this  in  such  a  way  that  one  half  of  the  total  loading 
of  the  entire  slab  is  carried  by  one  set  of  side  belts,  and  the  other 
half  by  a  second  set  which  crosses  the  first  at  right  angles. 


180  DEFLECTION   OF    SIDE    BELTS 

In  those  parts  of  the  slab  area  where  these  sets  of  belts  cross, 
forming  the  heading  of  the  columns,  the  loading  is  superposed  also. 

The  preceeding  investigation  of  the  shears  at  the  edges  of  side 
belts  and  their  loading  is  independent  of  their  width  and  of  the  posi- 
tion of  the  lines  of  contra-flexure,  but  their  width  will  be  assumed  in 
what  follows  to  be  determined  by  the  position  of  those  lines  as  shown 
in  (24)  on  account  of  the  independence  of  action  of  belts  of  their 
width,  as  previously  explained,  where  it  was  shown  that  no  bending 
moments  are  propagated  across  those  lines. 

The  question  now  arises,  how  the  vertical  shears  at  the  edges 
of  the  side  belts  are  distributed  across  their  width  and  carried  by 
them.  Since  by  symmetry  of  loading,  etc.,  there  is  no  vertical 
shear  at  the  edge  of  the  panel  where  y  =  b,  the  shear  must  diminish 
from  each  edge  of  a  belt  to  zero  at  that  line.  If  it  be  assumed  to 
diminish  uniformly,  that  is  equivalent  in  its  action  to  a  uniformly 
distributed  load  on  the  belt,  which  may  be  assumed  in  computation 
to  replace  the  shears  at  the  edges.  Whether  it  will  be  so  distributed 
or  not  depends  upon  the  stiffness  of  the  mushroom  head  and  the 
smallness  of  its  flexure.  Extensometer  measurements  on  the  rods 
of  the  side  belt  of  the  floor  slab  of  the  St.  Paul  Bread  Company 
Building  by  Prof.  Wm.  H.  Kavanaugh  show  beyond  question  that 
in  the  mushroom  system  the  load  is  so  distributed.  Other  exten- 
someter  measurements  to  which  the  writer  has  access  also  show  that 
in  systems  in  which  the  heading  of  the  column  is  not  so  stiff  as  this 
the  distribution  of  loading  cannot  be  taken  as  uniform  over  the  side 
belts. 

Now  the  belt  parallel  to  x  was  shown  to  carry  a  load  per  unit  of 
length  of  q  b  and  to  have  a  width  2(6  —  yi),  in  general,  or  a  width 
26(1  — J^3)  for  the  belt  between  the  lines  of  contraflexure;  hence 
the  intensity  of  the  loading  on  this  belt  is  q  6/2(6 — yi),  instead  of 
q,  as  it  would  be  in  a  uniformly  loaded  panel  duly  supported  at  its 
edges  by  beams  from  column  to  column.  Let  2 A,  designate  the 
area  of  the  effective  right  cross  section  of  the  steel  in  the  entire  width 
of  a  side  belt  regarded  as  forming  a  single  sheet  of  metal  of  the  width 
of  the  belt;  then  SA/2(6 —  y\)  is  the  effective  right  cross  section 
per  unit  of  width  of  belt,  and  we  may  write  (14)  in  the  form 

I  =  ij  d2  2 A  /  2(6  —  yi) (29) 

We  shall  consider  admissible  values  of  SA  later. 

Since  the  deflection  of  the  side  belts  may  be  taken  independently 
of  the  rest  of  the  slab,  let  those  values  for  the  intensity  of  loading 
and  the  moment  of  inertia  (29)  be  introduced  into  (21). 


STRESSES    IN    SIDE    BELTS  181 

We  then  obtain  an  expression  for  the  law  governing  the  deflec- 
tion of  that  part  of  the  side  belts  parallel  to  x  which  lies  between  the 
mushroom  heads,  and  is  bounded  by  lines  of  contra-flexure,  viz: 

z  =  ~~R2b  [  (a2  ~  x2)2  +  (b2  -  y2)2  ] (30) 


with  a  corresponding  equation  for  the  side  belts  parallel  to  y,  which 
may  be  obtained  by  replacing  q  b  in  (30)  by  q  a.  Call  this  second 
equation  (31).  Now  (30)  and  (31)  would  hold  thruout  the  entire 
length  of  these  belts  from  column  to  column  were  they  entirely 
separate  from  each  other  and  from  the  diagonal  belts  where  they 
cross  each  other.  It  will  be  necessary  later  to  obtain  the  equation 
which  holds  true  where  these  belts  cross  and  combine  with  each  other. 

9.  Practical  formulas  for  the  stresses  in  the  steel  and  concrete 
of  side  belts  between  the  lines  of  contra-flexure  will  now  be  obtained 
from  (30)  and  (31). 

In  order  to  do  this,  consider  the  summation  in  (30)  expressing 
the  effective  cross  section  of  the  steel  in  the  mid  area  of  the  side  belt 
regarded  as  forming  a  single  uniform  sheet,  that  mid  area  being 
bounded  on  all  sides  by  lines  of  contra-flexure. 

It  is  to  be  noticed  that  the  factor  (1  —  K2)  of  (30)  takes  into 
account  the  fact  that  the  lattice  of  rods  forming  the  reinforcement  is 
less  effective  than  the  same  amount  of  metal  in  the  form  of  a  sheet, 
the  only  question  left  being  this:  Will  the  great  irregularity  of 
distribution  of  the  reinforcement  in  this  area  cause  it  to  act  differ- 
ently to  any  noticeable  extent  from  the  manner  in  which  the  same 
amount  of  metal  would  act  were  it  possible  to  distribute  it  uniformly 
over  the  entire  area?  There  are  strong  reasons  which  go  to  sustain 
the  view  that  this  irregularity  of  distribution  is  negligible  in  the 
standard  mushroom  slab,  at  least  for  loads  less  than  those  that 
stress  the  steel  below  the  yield  point,  or  do  not  stress  the  concrete 
for  too  long  a  time  while  it  is  imperfectly  cured.  On  examining  a 
diagram  of  the  reinforcing  rods  of  a  slab  made  with  square  panels 
of  such  proportions  that  the  width  of  the  belts  is  one  half  the  distance 
between  columns,  then  the  pattern  previously  mentioned  into  which 
it  would  be  divided  by  these  belts  will  be  seen  to  consist  of  equal 
squares  whose  edges  are  equal  to  the  width  of  the  belts,  with  one 
central  square  in  each  panel  concave  upwards,  and  one  half  of  each 
of  the  saddle  shaped  squares  which  border  it,  also  lying  within  the 
same  panel,  and  one  quarter  of  each  of  the  four  convex  squares  at  the 


182  MEAN    REINFORCEMENT    OF    SIDE    BELT 

head  of  each  of  the  columns  at  the  corners  of  the  panel,  also  lying 
within  the  same  panel,  see  Fig.  55,  page  164. 

Each  side  square  will  be  found  in  this  case  to  have  double  (or 
two  belt)  reinforcement  over  one  half  or  its  area,  single  belt  rein- 
forcement over  a  diamond  occupying  one  fourth  of  its  mid  area,  and 
triple  reinforcement  over  four  triangular  areas  along  its  sides  which 
together  cover  one  fourth  of  the  square.  This  gives  a  mean  value 
of  S-A  =  2  A  i  in  which  AI  is  the  total  right  cross  section  of  the  rods 
in  the  side  belt. 

The  belts  in  the  standard  mushroom  are,  however,  not  so  wide 
as  this,  since  that  system  simply  requires  that  the  edges  of  the  side 
and  diagonal  belts  intersect  in  a  single  point,  Fig.  54,  instead  of  forming 
four  areas  of  triple  reinforcement  on  the  sides.  This  makes  the 
width  of  the  singly  reinforced  diamond  sufficient  to  just  reach  across 
the  side  belt.  In  this  practical  case  we  find  that  very  approximately 

SA  =  1.5  AI (32) 

in  which,  as  before,  AI  is  the  total  right  cross  section  of  the  side  belt 
in  square  inches.  It  is  evidently  impossible  for  this  single  side  belt 
of  rods  which  crosses  the  diamond,  to  elongate  without  a  correspond- 
ing equal  elongation  of  the  double  reinforcement  on  all  its  sides,  or 
at  least  it  is  impossible  for  readjustments  to  take  place  in  any  short 
time  such  as  will  make  these  direct  deformations  within  the  diamond 
larger  than  those  in  the  areas  along  side  of  it,  or  before  somewhat 
more  permanent  deformations  have  taken  place  in  the  concrete. 

In  cases  where  the  column  heads  are  smaller  than  the  standard, 
and  the  side  belts  still  narrower,  not  only  may  2A  become  much 
less  than  1.5  A  i  but  the  belt  become  so  weakened  near  the  central 
diamond  as  to  render  it  very  questionable  whether  the  irregularity 
of  distribution  of  steel  in  the  area  considered  may  be  safely  disre- 
garded. Diminution  of  the  size  of  the  heading  thus  not  only  dim- 
inishes cantilever  action,  but  reduces  the  effective  resistance  of  the 
reinforcing  steel.  Not  much  diminution  of  the  size  of  head  would 
be  required  to  reduce  the  value  of  S A  to  an  amount  as  small  as  At. 

Introducing  the  estimate  given  in  (32)  for  the  standard  mush- 
room into  (30)  we  derive  by  (23),  (23a)  and  (3),  for  that  part  of  the 
side  belt  parallel  to  x  between  x  =  +  %  aVs  and  x  =  —  J  a\/3, 

52  z  (1  —  K2)q  b 

/s  =  Ee,  =  ±  E  id-   -  =  ±  -(3x2  -  a2) 

5  x2  !SjdA1 


_  j£2\ 

q  b  (3x2  —  a2) 


12 


(33) 


TRUE    STRESSES    IN    SIDE    BELT  183 

in  which  MI  is  the  total  true  moment  of  resistance  of  the  side  belt, 
/s  is  the  true  stress  per  square  unit  of  the  reinforcement  in  the  side 
belt,  and  1.5  A  i  is  the  effective  right  cross  section  of  the  reinforce- 
ment. This  is  independent  of  y  as  before  noted,  showing  that  the 
values  of  /s  and  e\  are  the  same  for  one  rod  as  for  another,  but  they 
attain  their  greatest  values  at  the  mid  length  where  x  =  0.  If  units 
be  pounds  and  inches,  and  we  assume  j  =  0.91  for  the  very  small 
percentage  of  reinforcement  of  the  standard  mushroom  system,  then 
by  (33)  and  (6)  the  practical  formulas  for  design  are: 


/s     = 


3  q  a2  b  W  L 


4x  18x0.91^1  AX        175  di  A1 

W  L 

M1  =  1.5AXM/H  = 

128 


(34) 


in  which  /s  is  the  true  stress  in  the  steel,  and  MI  is  the  true  bending 
moment  of  the  effective  cross  section  1.5  A  i  of  the  steel  in  the  entire 
belt  as  shown  by  the  elongation  (at  mid  span)  of  the  rods  in  a  side 
belt  of  length  L,  where  L  is  either  2a  or  26,  and  W  =  4  q  a  b  is  the 
total  load  on  the  panel  in  pounds,  where  di  is  the  vertical  distance 
from  the  center  of  the  rods  in  the  single  belt  at  mid  span  to  the  top 
surface  of  the  slab. 

While  the  values  obtained  from  (34)  are  conservative  for  j  =  0.91, 
corresponding  to  a  percentage  of  reinforcement  for  one  belt  of  less 
than  0.25%,  (34)  should  be  regarded  merely  in  the  light  of  a  speci- 
men equation  for  that  percentage,  and  any  slab  where  the  percent- 
age differs  materially  from  that  assumed  value  should  be  submitted 
to  separate  computation  in  the  same  manner. 

Values  of  j  are  given  for  beams  by  Turnearue  and  Maurer  in 
their  "  Reinforced  Concrete  Construction,"  page  57,  for  different 
percentages  of  reinforcement  on  the  straight  line  theory,  which 
latter  is  now  accepted  usage.  As  already  stated,  standard  mush- 
room design  makes  the  percentage  of  reinforcement  for  warehouse 
floors  where  the  panels  are,  say  20'  x  20',  as  low  as  0.25%  or  less,  at 
the  middle  of  the  side  belts,  reckoned  on  the  beam  theory.  But  in 
heavier  and  larger  construction  it  may  reach  0.33%. 

We  have  taken  the  mean  available  steel  in  the  belt  as  1.5  A\, 
hence  the  mean  slab  reinforcement  will  not  be  less  than  1.5  x  0.23  = 
0.4%  in  the  side  belt  areas  between  lines  of  contra-flexure. 

In  case  we  assume  the  ratio  of  Es  for  steel  to  Ec  for  concrete  to 
be  15,  as  is  often  prescribed,  we  find  the  above  stated  value  of  j  as  a 


184  THEORETICAL    AND    EXPERIMENTAL    STRESSES    COMPARED 

good  mean  value,  which  will  be  less  in  cases  where  the  percentage 
of  steel  is  greater.  The  small  percentage  of  steel  and  great  relative 
thickness  of  concrete  is  one  of  the  distinguishing  features  of  the 
standard  mushroom  design. 

We  may  write  (34)  in  the  form: 

W  L  W  L 

fs  =  -  -}andMl=Aljdfs  =  ....  (34a) 

175  d,  A1  192 

in  which  M i  is  the  true  bending  moment  of  the  actual  cross  section 
AI  at  mid  belt.  We  have  written  this  modification  of  (34), not  for 
use  in  design,  but  merely  for  the  purpose  of  instituting  a  comparison 
with  empirical  formulas  obtained  by  Mr.  Turner  to  express  the 
results  of  numerous  tests  made  by  him.  On  pages  26  and  28  of  his 
" Concrete  Steel  Construction"  he  has  given  equations  expressing 
the  values  of  stresses  and  moments  in  mushroom  slabs  which  in  our 
notation  may  be  written  as  follows: — 

WL  W  L  W  L 

MI  =  A!  jdfs  =  -       -  ,  and  /s  =  -  .  (35) 

200  200  x  0.85  d  Al      170  d  Al 

in  which  he  has  assumed  0.85  as  a  mean  value  of  j. 

It  is  seen  that  equations  (34a),  obtained  from  rational  theory 
alone,  are  in  practical  agreement  with  (35),  which  were  deduced 
from  experimental  tests  of  mushroom  slabs,  where  the  numerical 
coefficient  introduced  is  entirely  empirical. 

As  will  be  seen  later,  (34)  is  the  equation  which  ultimately  con- 
trols the  design  of  the  slab  reinforcement;  so  that  the  agreement  of 
these  two  entirely  independent  methods  of  establishing  this  funda- 
mental equation  cannot  but  be  regarded  with  great  satisfaction  as 
affording  a  secure  basis  for  designs  that  may  be  safely  guaranteed 
by  the  constructor,  as  has  been  the  custom  in  constructing  standard 
mushroom  slabs. 

The  slab  theory  here  put  forth  diverges  so  radically  from  the 
results  of  beam  theory  that  we  introduce  here  the  following  compar- 
ative computation  of  the  smallest  values  of  true  bending  moment 
and  stress  in  steel,  which  can  be  obtained  by  beam  theory  for  the 
side  belt  parallel  to  x,  as  follows: — 

That  part  of  the  side  belt  between  the  lines  of  contra-flexure  is 
simply  supported  at  its  ends  by  shearing  stresses,  and  so  may  be 
taken  to  be  a  simple  beam  resting  on  supports  at  these  end  lines. 


STRESSES    BY    BEAM    THEORY    AND    BY    TEST  185 

Hence  the  true  stress  /s  and  the  true  bending  moment  M '  at  the 
middle  of  this  simple  uniformly  loaded  beam  may  be  computed  from 
the  equation, 

M'  =  A,jdfs  =  I  W'  L' ....(36) 

in  which  M'  is  the  total  moment  of  resistance. 

AI  is  the  total  right  cross  section  of  the  reinforcement,  W'  is  the 
total  uniformly  distributed  load,  and  L'  is  the  length  of  the  beam. 
The  length  of  the  simple  beam  in  that  case  is  evidently  the  distance 
along  x  between  lines  of  contra-flexure,  viz,  L  =  f  aV%  =  ^  L  V$, 
where  L  is  the  edge  of  the  panel,  and  the  total  load  at  most  will  be 
that  already  proven  to  be  carried  by  the  side  belt  viz,  q  b  per  unit  of 
length,  or  a  total  for  a  span  L'of  W'  =  qb  L'  =  %  qab  1/3  =  ^W  V% 
where  W  =  4  qab  is  the  total  load  on  the  panel,  hence 

W  L 

M'  =  A,jdfs  =  (36a) 

It  thus  appears  that  according  to  simple  beam  theory  the  true 
stress,  or  the  cross  section  of  steel  required  in  the  belt,  is  four  times 
that  obtained  by  slab  theory  as  shown  by  (34a).  Since  (34a)  is  in 
good  accord  with  experimental  tests,  this  comparison  justifies  the 
statements  made  near  the  beginning  of  this  paper  respecting  the 
inapplicability  of  beam  theory  to  the  computation  of  slab  design. 

The  floor  of  the  St.  Paul  Bread  Co.  Building,  previously  men- 
tioned,is  a  rough  slab  6"  thick,  and  has  panels  W  x  15',  with  ten 
3/8"  round  rod  reinforcement  in  each  belt,  built  for  a  design  load  of  100 
pounds  per  square  foot;  constructed  in  winter  and  frozen,  the  final 
test  was  not  made  until  the  end  of  its  first  summer  after  unusually 
complete  curing,  such  as  might  make  the  value  of  K  given  in  (6) 
not  entirely  applicable.  In  one  long  side  belt,  extensometer  measure- 
ments were  made  at  the  mid  span  on  three  rods,  (1)  a  middle  rod, 
(2)  an  intermediate  rod  and  (3)  an  outside  rod  of  the  belt,  with  the 
following  values  of  fs  in  pounds  per  square  inch  for  the  given  live 
load  in  pounds  per  square  foot: 

Live  Loads  108 . 4 1  316 . 8  #  416 . 8  # 


f,  =  E  c,  (1) 

(2) 
(3) 

7650 
7080 
7320 

15000 
14190 
13920 

17940 
16470 
17160 

Average 

7350 

14370 

17200 

A  by  (34) 

5000 

14440 

19000 

186  STRESSES   AT   YIELD    POINT 

The  observed  results  are  seen  to  be  in  excellent  agreement  with 
those  computed  from  (34)  for  the  heavier  loads,  while  any  disagree- 
ment is  on  the  safe  side.  Agreement  is  not  expected  for  light  loads. 

The  accuracy  and  applicability  of  (34)  and  preceeding  formulas 
is  dependent  on  the  fixity  of  the  lines  of  contra-flexure  (24)  which 
were  previously  stated  to  be  practically  immovable  because  of  the 
sudden  large  change  of  the  moment  of  resistance  of  the  slab  at  those 
lines.  That  fact  may  be  put  in  a  more  definite  and  convincing 
form  than  has  been  done  so  far.  Consider  for  a  moment  that  form 
of  continuous  cantilever  bridge  where  there  are  joints  between  the 
cantilevers  over  the  successive  piers  (which  are  in  the  form  of  a  letter 
T)  and  the  intermediate  short  spans  which  connect  the  extremities 
of  the  cantilevers.  At  such  joints  the  resisting  moments  vanish,  and 
they  form  in  a  sense  artificially  fixed  points  of  contra-flexure.  The 
same  thing  approximately  occurs  at  the  edge  of  the  mushroom, 
because  there  the  reinforcing  steel  rapidly  dips  down  from  a  level 
above  the  neutral  plane  to  one  below  it,  and  the  sign  of  the  moment 
of  resistance  changes  thru  zero  at  that  edge. 

Furthermore,  it  may  be  proper  to  state  in  this  connection  that 
the  foregoing  theory  has  been  developed  in  consonance  with  the 
general  principles  of  elasticity,  and  that  somewhat  different  condi- 
tions and  relations  are  thought  to  exist  when  the  steel  at  the  middle 
of  the  side  belts  reaches  its  yield  point,  as  it  does  in  advance  of  the 
rest  of  the  reinforcement.  As  the  yield  point  is  reached  equations 
(34)  no  longer  hold;  for,  as  will  be  seen  more  clearly  later,  the  single 
belt  of  reinforcing  steel,  which  crosses  the  circumference  of  an  ap- 
proximately circular  area  of  radius  L  /  2  about  the  center  of  each 
column,  will  everywhere  reach  the  yield  point  at  practically  the  same 
instant,  and  if  loaded  much  beyond  this  will  develop  a  continuous 
line  of  weakness  there.  The  equations  that  hold  in  this  case  will  be 
approximately  those  due  to  the  actual  cross  section  A  i  of  the  belt,  in 
place  of  (34),  which  contain  the  effective  cross  section,  viz: 

3  q  a2  b  W  L 

4  x  12  x  0.91  d,  A^        117  d, 

(37) 

W  L 
Ml  =  A1jdlfs  =  - 

128 

which  may  be  regarded  as  expressing  the  relations  that  exist  at  the 
limit  of  the  elastic  strength  of  the  slab  and  the  beginning  of  perma- 
nent deformation,  tho  not  necessarily  of  collapse. 


STRESSES    IN    CONCRETE  187 

The  percentage  of  reinforcement  in  standard  mushroom  slabs 
is  small  enough  to  make  their  elastic  properties  depend  upon 
the  resistance  of  the  steel.  The  stresses  in  the  concrete  may  then  be 
be  computed  from  those  in  the  steel,  but  many  uncertainties  attend 
any  such  computation.  It  is  usage,  fixed  by  the  ordinances  of  the 
building  codes  of  most  cities  to  require  the  application  of  the  so 
called  " straight  line  theory"  in  such  computations,  not  because  that 
will  give  results  which  will  be  verified  by  extensometer  tests  of  com- 
pressions in  the  concrete,  for  it  will  not,  but  because  it  is  definite 
and  on  the  side  of  safety.  Furthermore  it  is  usually  prescribed 
that  the  ratio  of  the  modulus  of  elasticity  of  steel  divided  by  that 
of  concrete  shall  be  assumed  to  be  15,  where  the  moduli  are  unknown 
by  actual  test  of  the  materials.  This  is  usually  far  from  a  correct 
value.  The  consequence  is  that  the  results  of  computation  of  the 
stresses  in  concrete  are  highly  artificial  in  character,  and  should  not 
be  expected  to  be  in  agreement  with  extensometer  tests.  With  this 
understanding  the  computed  stress  in  the  concrete  at  the  middle  of 
the  side  belt  will  be  found  as  follows: 

Let  id  be  the  distance  from  the  center  of  the  steel  to  the  neutral 
plane.  (It  happens  to  be  more  convenient  in  this  investigation  to 
use  this  distance  id  here  and  in  our  previous  formulas  than  to  intro- 
duce the  distance  from  the  neutral  axis  of  the  slab  to  the  compressed 
surface  of  the  concrete,  as  is  done  by  many  writers,  under  the  desig- 
nation k  d.  These  quantities  are  so  related  that  i  +  k  =  1  ). 

Then,  as  is  well  known  from  the  geometry  of  the  flexure  of 
reinforced  concrete  beams,  in  case  tension  of  concrete  is  disregarded, 

7  77* 

/,  =  JL--C/S (38) 

i      E, 


where   the   subscripts  c    and    s    refer    to    concrete    and    to    steel 
respectively. 

Applying  (38)  to  the  greatest  computed  stress  fs  =  19000  in 
the  St.  Paul  Bread  Go's  Building,  gives  a  computed  stress  fc  =  492; 
but  taking  the  greatest  observed  stress  /„  =  17940  gives  fc  =  465  Ibs. 
per  sq.  inch,  as  the  greatest  computed  compressive  stress  in  the 
concrete  at  the  middle  of  the  side  belt,  if  i  =  0.72. 

The  tensile  stress  across  the  middle  of  the  side  belt  at  the 
extreme  fiber  of  its  upper  surface  is  fixed  by  the  curvature  of  the 
vertical  sections  of  the  slab  in  planes  that  cut  the  side  belt  at  right 
angles.  As  stated  previously  all  such  planes  make  cross  sections  of 
the  side  belt  that  are  identical  in  shape.  That  is  a  consequence  of 


188  DEFLECTIONS    IN    COLTTMN    HEAD    AREAS 

the  conclusion  reached  previously,  that  all  the  rods  in  the  side  belt 
are  subjected  to  equal  tensions.  The  curvature  of  these  sections 
is  controlled  by  the  stiffness  of  the  mushroom  heads,  which  is  so 
great  as  to  make  the  curvature  very  small.  No  considerable  tensile 
cross  stresses  are  consequently  to  be  apprehended;  but  in  case  the 
stiffness  of  the  head  were  to  be  decreased,  stresses  might  arise  such 
as  to  develop  longitudinal  cracks  over  the  middle  rod  of  the  side  belts. 

10.  In  order  to  obtain  practical  formulas  for  the  deflections  and 
stresses  in  the  steel  thruout  the  areas  at  and  near  the  tops  of  the 
columns  where  all  the  belts  cross  each  other,  and  lying  between  lines 
of  contra-flexure,  we  shall  have  recourse  to  (30)  and  (31)  which  are 
here  superimposed  on  each  other,  and  combined  together.  Were 
there  no  steel  here  in  addition  to  the  side  belts,  that  superposition 
could  be  correctly  effected  by  writing  a  value  of  z  whose  numerator 
would  be  the  sum  of  the  numerators  of  (30)  and  (31),  for  that  would 
superpose  the  loads  of  the  two  side  belts,  and  thus  place  the  total 
required  loading  upon  this  area  as  previously  explained;  and  then  by 
writing  for  a  denominator  the  sum  of  the  denominators  of  (30)  and 
(31),  for  that  would  superpose  and  combine  the  resistance  of  all  the 
steel  in  both  belts.  But  such  a  result  would  leave  out  of  account  the 
reinforcement  arising  from  the  diagonal  rods,  and  the  radial  and 
ring  rods,  which  should  also  be  reckoned  in  as  furnishing  part  of  the 
resistance. 

Supposing  this  additional  steel  to  be  distributed  in  this  area  in 
the  same  manner  as  is  that  of  the  side  belts,  a  supposition  which  is 
very  close  to  the  fact,  we  may  write 

-  2 


&2_ 


48Eijd2  A 

in  which  "2  A  is  the  cross  section  of  the  total  reinforcement  in  this 
area  regarded  as  forming  a  uniform  sheet,  i  and  j  stand  for  mean 
values  that  have  to  be  determined  by  the  percentage  of  reinforce- 
ment and  its  position,  while  d  is  the  mean  distance  of  the  center  of 
action  of  the  steel  above  the  lower  compressed  surface  of  the  con- 
crete at  the  point  xy. 

We  may  conservatively  assume  in  the  standard  mushroom 
that  the  center  of  action  of  the  steel  is  at  the  center  of  the  third  layer 
of  rods  from  the  top,  as  will  appear  more  clearly  later.  This  defines 
d,  which  we  shall  consequently  designate  by  ds. 

It  remains  therefore  to  estimate  the  amount  of  the  total  rein- 
forcement S  A,  and  then  find  mean  values  of  i  and  j. 


MEAN    REINFORCEMENT    OF    HEAD  189 

In  case  of  reinforcing  rods  which  are  all  of  them  continuous 
over  a  head  without  laps,  the  percentage  of  reinforcement  falls  only 
slightly  below  4  times  that  at  the  middle  of  a  side  belt;  but  on  the 
other  hand  were  none  of  them  continuous  for  more  than  one  panel 
and  each  lap  reached  beyond  the  center  of  the  column  to  the  edge 
of  the  mushroom,  the  percentage  of  reinforcement  would  not  be  less 
than  7  times  that  at  the  middle  of  a  side  belt,  and  to  this  must  be 
added  that  due  to  the  steel  in  the  radial  and  ring  rods.  Thus  the 
percentage  of  reinforcement  here  may  be  varied  not  only  by  reason 
of  the  larger  or  smaller  number  of  laps  over  each  mushroom,  but  by 
reason  of  the  length  of  the  laps,  from  perhaps  3.75  to  7  times  that 
at  the  middle  of  a  side  belt.  For  standard  mushroom  construction 
using  long  rods,  it  may  be  taken  conservatively  as  a  4.25  times  that 
at  the  middle  of  a  side  belt. 

It  is  impossible  to  make  an  estimate  that  will  be  accurate  for 
all  cases,  but  commonly  the  8  radial  rods  of  a  20'  x  20'  panel  are 
equivalent  in  amount  to  a  single  If"  round  rod,  or  a  I"  square 
bar  circumscribing  the  area  under  consideration,  that  is  to  4  square 
inches  of  additional  reinforcement  to  be  distributed  in  the  width  of 
a  single  side  belt. 

The  two  rings  rod,  of  which  the  larger  is  commonly  7/8"  round, 
and  the  smaller  5/8"  round,  may  be  taken  to  increase  the  reinforce- 
ment of  this  area  by  at  least  one  square  inch  of  cross  section,  giving 
all  told  some  five  square  inches  of  cross  section  additional,  equiva- 
lent forty-five  3/8"  round  rods,  or  twenty-one  1/2"  rods.  It  thus 
appears  that  the  increased  reinforcement  from  this  source  reaches 
from  2  to  4  times  Ait  and  we  may  safely  assume  a  mean  total  rein- 
forcement over  this  area  of 

SA  =  7.5  A! (40) 

of  which  the  center  of  action  may  be  pretty  accurately  stated  to  be 
at  the  middle  of  the  third  layer  of  reinforcement  rods  from  the  top. 

In  the  standard  design  of  mushroom  floors  for  warehouses  with 
panels  about  20'  x  20',  the  mean  percentage  of  reinforcement  for  a 
single  belt  AI  being  about  0.23%,  may  be  taken  by  (40)  for  a  rein- 
forcement 7.5  A!  as 

7.5  x  0.23  +   =  1.75%     The  corresponding  value  of  j  is  0.83, 

and  we  shall  have 

j  S  A  =  0.83  x  7.5  AI  =  6^1 (41) 

As  previously  stated,  these  equations  (containing  estimated  mean 
numerical  values)  are  given  as  a  specimen  computation  for  the  purpose 


190  STRESSES    AT   EDGE    OF    CAP 

of  making  comparisons.  In  actual  design,  computations  like  these 
should  be  made  which  introduce  the  exact  values  appearing  in  the 
design  under  consideration. 

We  now  derive  from  (39)  and  (40)  by  the  help  of  (23)  the  follow- 
ing equations  for  this  area  where  the  belts  all  cross : — 

52  z      ±(1  —  K2)  q  (a  +  6) 

^3  ~~    ;  =  ~~ 

ox  90  7  d3  A  i 

.(42) 

Q     ]££\ 

=  7.5  Al  jd3fK  =  -  ~q(a  +  b)  (3x2  —  a2) 

12 

in  which  j  and  d3  are  less  than  in  (33)  and  (34),  as  has  been  stated 
previously. 

Apply  (42)  to  find  the  stresses  at  the  edge  of  the  column  cap 
on  the  long  side  LI. 

Let  B  =  2x  be  the  shortest  distance  along  the  middle  of  the 
side  belt  parallel  to  x  between  the  edges  of  the  caps  of  two  adjacent 
columns,  and  introduce  the  values  j  =  0.83,  K  =  0.5,  and  W  =  4qab, 
then; 

W  LI  (Li  +  La)  (3B2/Ll  —1  ) 
800  d3A1L2 

W  L!  (L!  +  L2)  (  3B2/Ll  —  1  ) 


MI  =  7.5Aljd3fa  = 


128  L 


(43) 


in  which  7.5  AI  is  the  effective  cross  section  of  the  steel  in  this  area, 
and  MI  is  the  true  resisting  moment  of  the  total  steel  derived 
from  the  elongation,  and  d3  is  as  stated  after  (39). 

It  has  been  found  that  the  foregoing  theoretical  expressions  which 
neglect  to  take  account  of  the  local  stresses  induced  in  the  slab  just 
outside  the  edge  of  the  cap  by  reason  of  the  rigidity  of  the  cap  it- 
self are  incapable  of  giving  results  in  accordance  with  experimental 
data. 

Equation  (43)  implicitly  assumes  that  the  slab  while  supported 
on  the  cap  is  nevertheless  so  separate  and  independent  of  it  as  to 
have  no  great  rigidity  over  the  cap  than  elsewhere  and  is  conse- 
quently unaffected  by  the  mass  of  the  cap.  But  this  assumption 
is  not  in  accordance  with  the  fact  because  the  cap  is  integral  with 
the  slab  and  forms  a  nearly  rigid  boss  on  the  slab  which  largely 
prevents  bending  and  stretching  over  the  cap  so  that  inside  the 


GREATEST   STRESSES    OVER    COLUMN  191 

edge  of  the  cap  much  smaller  extensions  occur  in  the  belt  rods  than 
would  otherwise  occur,  and  less  than  do  occur  in  parallel  rods  out- 
side the  cap.  But  the  total  extensions  of  the  several  rods  of  a  belt 
from  one  line  of  inflection  to  another  must  accompany  one  another, 
and  those  rods  that  cross  the  cap  and  have  their  extensions  mostly 
prevented  inside  the  cap  must  suffer  correspondingly  greater  exten- 
sions in  the  remaining  parts  of  their  lengths  by  way  of  compensation 
for  this  loss.  This  effect  will  be  most  accentuated  just  where  the 
rods  cross  the  edges  of  the  cap  and  will  cause  abnormal  local  exten- 
sions especially  in  the  rods  along  the  middle  of  the  belts  that  cross 
the  edge  of  the  cap  nearly  perpendicularly.  It  is  just  here  that  the 
greatest  stresses  are  observed  as  might  be  expected.  The  stresses 
in  rods  tangent  to  the  cap  are  less  than  these,  and  in  the  other  rods 
which  are  nearer  the  edges  of  the  belts  the  stresses  are  smaller  also, 
partly  by  reason  of  the  low  level  at  which  they  are  usually  placed. 
These  circumstances  all  conspire  to  accentuate  the  stresses  of  the 
rods  at  the  edge  of  the  cap. 

It  is  not  possible  to  make  an  exact  mathematical  analysis  of  the 
resulting  abnormal  stresses.  But  the  form  of  (43)  has  suggested 
an  expression  which  will  enable  us  to  fix  a  limiting  value  to  those 
stresses  with  considerable  assurance,  since  the  greatest  steel  stresses 
which  have  been  observed  around  the  cap  in  extensometer  tests 
do  not  exceed  the  values  thus  computed. 

We  feel  sure  that  ample  allowance  for  this  increased  stress  is 
included  in  the  following  amended  expression  for  the  greatest  stresses 
in  the  side  belts  at  the  edge  of  the  cap: 

W  L!  (L+  LJ&Ll/B2  -  1) 


800*  Ax  L, 

WL,  (L,  +  L2) 
Jfx-T.SAxj*/.-  "128 

This  equation  gives  the  same  values  of  /s  and  M1  as  (43)  at  the  column 
center,  i.  e.  when  B  =  L^  But  at  other  points  (43)  and  (44)  diverge 
from  each  other,  for  at  any  points  between  the  column  center  and 
the  line  of  inflection  (43)  gives  smaller  values  of  /s  and  MI  than  at 
the  column  center,  but  (44)  gives  larger  values  for  the  stresses,  and 
thus  makes  allowance  for  abnormal  stresses.  Moreover  (44)  makes 
the  stresses  greater  the  smaller  B  is  and  the  larger  the  cap.  It  is 
evident  that  these  abnormal  stresses  should  increase  with  the  size 
of  the  cap  which  causes  them.  But  it  should  be  stated  again  and 
emphasized  that  the  values  thus  obtained  are  an  outside  limit,  and 
that  (44)  may  give  values  considerably  in  excess  of  observed  stresses. 


192  COMPRESSION    IN    CONCRETE    AROUND    CAP 

(44)  consequently  should  not  be  used  to  compute  the  amount  of 
reinforcement  required  in  any  proposed  design.  But  with  a  given 
design  it  is  possible  to  say  with  considerable  confidence  that  the 
greatest  stress  will  not  exceed  those  computed  by  (44). 

It  should  be  further  stated  in  this  connection  that  abnormal 
stresses  in  a  few  rods  at  the  center  of  a  belt  at  the  edge  of  the  cap 
is  a  local  phenomenon  of  no  serious  import  for  the  stability  of  the 
slab  because  such  stresses  if  sufficient  will  cause  a  slight  yielding 
at  the  poiat  which  will  bring  the  other  parallel  slab  rods  into  play 
to  assist  them. 

In  the  same  way  as  (44)  has  been  obtained  from  (43)  and  (43) 
from  (42),  a  similar  expression  may  be  obtained  for  the  abnormal 
stress  in  the  rods  of  diagonal  belts  at  the  edge  of  the  cap,*  but  the 
numerical  values  thus  obtained  differ  little  from  those  resulting  from 
(44).  This  expression  has  therefore  been  omitted  as  unimportant. 

Tests  and  long  experience  show  that  much  higher  compressive 
stresses  may  be  safely  permitted  in  the  concrete  around  column 
caps  where  the  compression  is  in  two  directions  at  once,  circumfer- 
ential and  radial,  than  in  ordinary  direct  one  way  compression. 
Aside  from  this  additional  resistance  to  these  converging  compres- 
sions the  Joint  Committee  has  recognized  a  greater  capacity  for 
resistance  to  compression  at  supports  than  elsewhere  in  recommend- 
ing that  in  general  "the  extreme  fiber  stress  of  a  beam  may  be 
allowed  to  reach  32.5  percent  of  the  compressive  strength,"  while 
" adjacent  to  the  support  of  continuous  beams  stresses  15  percent 
higher  may  be  used.'J 

If  we  use  no  higher  stresses  around  the  cap  than  those  allowed 
by  this  recommendation  for  beams  we  obtain  for  a  concrete  whose 
compressive  strength  is  taken  as  2000  Ibs.  per  square  inch  a  work- 
ing stress  of  2000  x  .325  x  1.15  =  747.5  Ibs.,  or  822  Ibs.  for  a  con- 
crete having  a  strength  of  2200  Ibs. 

Now  various  published  and  unpublished  extensometer  tests 
of  slabs  show  that  the  unit  deformation  in  the  concrete  immediately 
adjacent  to  the  edge  of  the  cap  designated  by  ec  has  a  mean  value 
of  not  more  than  seven  tenths  (0.7)  of  es  the  unit  deformation  of  the 
steel  in  the  top  'belt  immediately  above  it,  this  ratio  varying  under 
the  heavier  loads  between  0.5  and  0.85.  Hence  the  working  steel 
stress  /s  corresponding  to  a  working  stress  in  the  concrete  of 
/c  =  747.5  Ibs.  per  square  inch  is 


Son  Proc.  Am.  Soc.  C.  E.  Jan.  1914,  p.  77. 


STRESSES    OVER    CAP  193 


=     s^fc  =  15  x  747.5/0.7  =  16000  .................  (45) 

E 


'a 


Ibs.  per  square  inch. 

It  thus  appears  that  ordinary  working  stresses  in  the  steel  at  the  edge 
of  the  cap  will  be  accompanied  by  safe  stresses  in  the  concrete.  As 
a  matter  of  experience,  it  has  known  that  failure  arising  from  com- 
pression around  the  column  is  practically  unknown  in  concrete  that 
has  had  opportunity  to  become  reasonably  hard. 

It  will  be  noticed  that  in  order  to  make  fs  and  fc  as  small  as 
possible  in  this  area  d3  must  be  made  as  large  as  possible,  i.  e.,  the 
steel  at  the  edge  of  the  cap  must  be  raised  as  near  the  top  of  the  slab 
as  possible.  Neglect  of  this  is  to  invite  failure  and  weakness  such 
as  has  overtaken  certain  imitators  of  the  mushroom  system. 

A  final  remark  is  here  in  place  respecting  the  values  of  j  and  d% 
in  this  area.  The  stresses  /s  and  /c  diminish  very  rapidly  towards 
the  lines  of  contra-flexure,  where  they  vanish,  and  the  fact  that  the 
steel  also  rapidly  increases  its  distance  from  the  top  of  the  slab  at 
the  same  time  might  be  regarded  at  first  thought  as  requiring  some 
modification  of  the  assumptions  we  have  made  as  to  the  values  of 
j  and  c?3,  which  are  approximately  correct  at  the  edge  of  the  cap 
where  the  steel  is  placed  as  near  the  top  surface  as  due  covering  will 
permit.  But  the  fact  is  this:  the  only  consideration  of  importance 
is  the  one  respecting  the  position  of  the  steel  in  that  part  of  this 
area  where  the  moments  and  stresses  are  large.  The  effect  of  the 
position  of  the  steel  near  the  lines  of  contra-flexure  is  negligible,  and 
the  fact  that  the  amount  of  reinforcement  may  be  somewhat  smaller 
near  these  lines  than  elsewhere  may  also  be  neglected,  so  that  the 
mean  effective  reinforcement  previously  estimated  is  likely  to  be  an 
underestimate  rather  than  the  reverse.  Further,  the  fact  that  the 
slab  is  practically  clamped  horizontally  either  at  the  edge  of  the  cap 
or  the  edge  of  the  superposed  column,  instead  of  at  its  center  as 
assumed  in  our  formulas,  renders  the  results  given  thus  far  slightly 
too  large. 

Good  average  values  of  the  size  of  steel  used  in  the  standard 
mushroom  system  of  medium  span  would  make  the  radial  rods 
9/8"  round,  the  outer  ring  rod  7/8"  round,  the  inner  ring  rod 
5/8"  and  the  belt  rods  3/8"  round.  The  importance  of  having 
the  belt  rods  small  is  that  for  a  given  thickness  of  slab  the  smaller 
these  rods  are  the  larger  is  d  in  both  (34)  and  (43)  and  consequently 
the  smaller  is/s  and  A\. 

11.     In  attempting  to  consider  the  stresses  in  the  diagonal  rods 
of  the  central  rectangle  between  the  side  belts  of  a  panel,  it  will  be 


194  DEFLECTIONS  IN  CENTRAL  AREA  OF  PANEL 

noticed,  as  stated  before,  that  no  true  bending  moments  are  propo- 
gated  across  the  vertical  planes  or  lines  of  contra-flexure  (24)  which 
bound  it,  and  since  the  vertical  shearing  stresses  at  these  lines  are 
uniformly  distributed  along  them,  as  already  shown,  (28),  there  are 
no  true  twisting  moments  in  these  planes.  The  curvatures  of  this 
rectangle  will  consequently  depend  upon  its  own  loading  and  the 
resistance  of  its  own  moment  of  inertia,  regarded  as  uniformly  dis- 
tributed, independently  of  that  of  other  parts  of  the  slab. 

Hence  (21)  may  be  correctly  applied  to  this  area,  regardless  of 
the  values  which  /  (and  q)  may  assume  elsewhere,  provided  only 
that  the  values  of  /  in  other  areas  may  be  assumed  to  have  constant 
values  thruout  those  areas,  and,  further,  that  those  areas  are  sym- 
metrically disposed,  so  that  all  central  rectangles  have  one  and  the 
same  given  value  of  /  thruout,  all  side  belts  also  have  one  given 
value  of  7,  and  the  mushroom  heads  have  a  given  value  also,  each  of 
these  three  sorts  of  areas  being  independent.  The  truth  of  this 
proposition  has  been  heretofore  tacitly  assumed  in  applying  (21) 
to  these  latter  areas  as  has  been  done. 

It  will  be  seen  however,  that  the  values  of  z  obtained  from  such 
diverse  equations  express  deflections  of  any  point  xy  on  the  supposi- 
tion that  all  the  areas  considered  have  the  same  value  of  7;  but  these 
separate  equations,  each  with  its  own  peculiar  value  of  7,  can  be 
used  separately  to  find  the  difference  of  level  zl  —  z2  between  any 
two  points  Xi  y\  and  x2  y2  which  lie  in  an  area  where  7  may  be  regarded 
as  constant.  We  shall  return  to  this  point  when  we  come  to  the 
derivation  of  practical  deflection  formulas. 

For  convenience  in  computing  stresses  in  the  rods  of  the  diago- 
nal belt,  let  the  direction  of  the  coordinates  be  changed  so  that  in 
square  panels  they  will  lie  along  the  diagonals  which  make  angles 
of  45  °  with  those  used  thus  far.  In  (21)  let 

x  =  %V~2(x'  +  y'),         y  =  ^2(x'  --y'),  then 

V  '      ^*     *7     I"      J  O^'O  '9v  '*>      >9  i    /      '9     .          '9\  91  s  A  te\ 

z=  -  -  [a4-~a2(x2  -f-  y2)  +  x2y2  +  \(x 2-{-y 2) 2].  .(47) 

24  E  i  j  d22A 

in  which  the  panel  is  square  and  the  axes  of  x'  and  y'  lie  along  its 
diagonals,  while  the  value  of  2,A/g  is  the  effective  cross  section  per 
unit  of  width  of  all  the  reinforcement  in  this  area  regarded  as  a 
single  uniform  sheet  of  metal,  and  g  =  7/8  a,  is  the  width  of  a 
diagonal  belt,  and  is  equal  to  the  diameter  of  the  mushroom  head. 
In  rectangular  panels  g  =  7/16  (a  +  b). 


ELONGATIONS    AND    SHEARS    IN    CENTRAL    AREA  195 

From  (34)  we  have 

24  #  ij  d22A 

d2  z  52z        (1  —K2)qg 

'  •       ^  •    i  *  /     I    i7  9  t.) 


d2z         (l—K2)qgx'y' 

and  - (50) 

8x  by          ±Eijd22A 

These  expressions  satisfy  (20)  as  they  should,  for  (20)  is  inde- 
pendent of  the  directions  of  the  rectangular  axes  x  and  y. 

From  (49)  it  appears  that  e\  =  0  =  /s,  on  the  circumference  of 
the  circle  x'2-\-y'2  =  fa2,  which  passes  thru  the  points  where  the 
lines  of  contra-flexure  intersect. 

By  (19),  which  holds  for  any  rectangular  axes,  and  by  (50), 
we  find 

n'  =  l(i—K)qx'y'.  ..(26)' 

From  (26) '  it  appears  that  n  sections  by  all  vertical  planes 
parallel  to  the  diagonals,  the  twisting  increases  uniformly  with  the 
distance  from  the  diagonal. 

Hence  by  (9)  we  have 


-(^ 

\dx 


\ ux  uy   / 

.(28) 


(dm2       Sn '  \ 
v+^ 


So 


It  thus  appears  that  the  same  law  holds  for  vertical  shearing 
stresses  on  planes  parallel  to  the  diagonals,  as  holds  in  (28)  for  planes 
parallel  to  the  edges  of  the  panel. 

In  standard  mushroom  designs  the  edges  of  the  diagonal  belts 
intersect  on  or  very  near  to  the  edges  of  the  side  belts.  That  makes 
the  middle  half  of  the  central  square  to  be  covered  by  double  belting, 
and  the  remainder  of  it  by  single  belting,  so  that  2 A  =  1.5A2  or 
perhaps  1.6  A2,  and  the  mean  value  of  A,  the  reinforcement  per 
unit  of  width  of  slab  here,  is  to  be  found  by  dividing  this  by  the 
width  of  a  belt,  which  is  7/8  a.  We  should  then  find  A  =  1.5  A2/ 
7/8  a  =  1.7  A2/  a.  But  this  mean  value  of  A  is  not  its  mean  effect- 
ive value  for  this  area,  because  the  reinforcement  is  so  disposed  as 


196  MEAN  REINFORCEMENT  STRESS  AT  PANEL  CENTER 

to  furnish  the  larger  values  of  /  in  the  central  diamond  just  where 
the  largest  true  applied  moments  and  stresses  occur.  The  mean 
value  of  A  in  the  central  diamond  is  2^42/7/8a  =  2.3^42/a.  The 
mean  effective  value  lies  between  these  two  extremes,  probably 
nearer  the  latter  than  the  former.  A  similar  question  was  discussed 
in  connection  with  (40)  and  (41).  We  shall  assume  as  the  mean 
effective  reinforcement  in  this  central  rectangle, 

A  =  2A2/a,  and  I  =  2A2  ij  d\/a 
or  in  case  of  rectangular  panels 

7  =  4A2  ij  dl/  (a  +  6) (51) 

In  case  of  rectangular  panels  the  term  2a2  in  (49)  should  be  replaced 
by  a2  +  b2  as  a  mean  value  to  make  it  depend  the  dimensions  of  the 
panel  symmetrically,  as  it  must.  Making  these  substitutions  in 
(49)  we  have  at  x  =  0  =  y'  the  center  of  the  panel. 

W  (Li  +  L2)  (L?  +  Z|)         Ci  W  Li 
/s  =  Ee'  =  - 

1024  L!  L2  A2  j  d2  256  A2  j  d2 

W(T          T^T^    rt       r    WT    * (52) 

W (L/i  -f-  L2)  (JLi  -f-  L2)        LI  W  L 

Mi  =  2A2  j  d2fs  =  - 

512  L!  L2  128 

where  d  =  \(Ll/L2  +!)(!+  L22/Li2).     Take  j  =  0.89. 

If  1  >  L2 /I/!  >  0.75  then  1  <  Ci  <  1.042,  hence  Cl  varies  less  than 5% 
while  Z/2/Z/i  varies  by  25%  between  its  extreme  permissible  values. 
Ci  may  ordinarily  be  taken  as  unity,  or  may  be  found  with  sufficient 
precision  by  interpolation  between  the  values  just  given. 

The  steps  by  which  these  equations  (52)  were  deduced  may  not 
seem  conclusive,  since  they  are  not  rigorous.  They  need  be  only 
good,  working  approximations  for  the  purpose  for  which  they  will  be 
here  used,  viz,  to  show  that  the  stresses  at  the  center  of  the  panel 
are  less  than  those  at  the  mid  span  of  the  side  belts  in  case  A  i  =  A2. 

The  value  of  d2  in  (52)  is  less  than  di  in  (34),  but  always  more 
than  90%  of  it.  We  may  define  d2  as  the  vertical  distance  from  the 
center  of  the  second  and  upper  of  the  two  diagonal  belts  to  the  top  sur- 
face of  the  concrete.  We  may  assume  d2  =  0.9^  and  j  =  0.89  in 
(52),  and  then  we  may  compare  these  stresses  for  a  square  panel  as 
follows : — 

175 

/.    =  --/. (53) 

205 

where  /s  refers  to  the  center  of  the  panel.      Even  were  the  smaller 


LINE    OF   ULTIMATE   WEAKNESS 


value  for  the  mean  reinforcement,  1.7  A2/ a,  used  in  deriving  (52) 
and  (53),  the  stress  given  by  these  equations  would  not  exceed  that 
given  by  (34).  The  compressive  stress  /c  in  the  concrete  at  the 
center  of  the  panel  may  readily  reach  a  dangerous  value  in  case  the 
forms  are  removed  too  soon.  It  should  therefore  be  carefully  con- 
sidered in  each  case.  Here,  we  have  an  approximate  value  of  i  =  2/3 
and  (38)  then  becomes /c  =  /s/30  with  no  possible  assistance  from 
steel  reinforcement  since  that  is  all  on  the  bottom  of  the  slab.  An 
estimate  that  the  elastic  stress  in  the  steel  at  the  center  of  the  panel 
does  not  much  exceed  80%  of  that  at  the  middle  of  the  side  belt 
cannot  be  far  from  the  truth. 

While  this  is  undoubtedly  the  fact,  it  will  appear  on  further 
consideration  that  local  stresses  and  strains  which  exist  at  incipient 
failure  are  of  such  magnitude  as  to  ma-ke  the  weakest  points  of  the 
diagonal  belts  to  lie  ultimately  not  at  the  center,  but,  instead,  just 
outside  the  diamond  where  they  cross  each  other. 

Take  the  standard  case  where  the  central  diamond  reaches  just 
across  to  the  side  belts.  For  square  panels  imagine  a  circle  to  be 
drawn  concentric  with  each  column  of  radius  L/2.  Any  circle  at  a 
column  will  be  tangent  to  the  edges  of  four  diagonal  belts  across  the 
tops  of  the  four  columns  adjacent  to  it,  and  then  the  octagon  cir- 
cumscribing it,  whose  sides  cut  at  right  angles  all  the  belts  that  cross 
this  column  head,  intersects  but  a  single  belt  of  rods  as  every  point 
of  its  perimeter.  It  is  evident  that,  so  far  as  reinforcement  is  con- 
cerned, such  a  line  or  section  cuts  less  steel  per  unit  of  perimeter 
than  any  other  regular  figure  concentric  with  the  column  and  that 
the  reinforcement  is  entirely  symmetrically  disposed  about  the 
column  center,  so  that  in  case  of  equal  diagonal  and  side  belts,  it 
would  be  impossible  from  their  geometry  to  distinguish  the  one  from 
the  other  by  anything  inside  the  octagon.  That  fact  would  make  it 
inherently  probably  that  the  stresses  and  strains  of  the  rods  where 
they  cross  any  one  side  of  this  octagon  should  be  approximately  the 
same  ultimately  as  in  those  that  cross  any  other  side,  whether  they 
be  rods  in  a  diagonal  belt  or  in  a  side  belt.  And  what  will  be  at- 
tempted to  be  shown  immediately  is  that  ultimately  the  stresses 
and  strains  in  these  several  belts  approach  equality.  If  that  should 
be  established,  it  will  follow  from  the  conclusion  already  reached  as 
to  the  excess  of  the  stresses  and  strains  of  the  side  belt  over  those  at 
the  center  of  the  panel,  that  ultimately  those  at  the  edges  of  the 
ocatgon  exceed  those  in  the  same  rods  at  the  center  of  the  panel. 

The  qualification  implied  above  in  affirming  that  this  is  what 
will  occur  ultimately,  is  for  the  purpose  of  conveying  the  idea  that 


198  LOCAL  STRESSES  AND  STRAINS 

this  is  the  approximate  distribution  of  stresses  and  strains  which 
will  take  place  when  the  slab  is  sufficiently  loaded  to  bring  the  steel 
at  the  middle  of  the  side  belt  to  the  yield  point.  At  less  stress  than 
this  there  is  so  much  lag  in  the  distribution  of  the  effect  of  loading 
that  it  penetrates  to  the  various  parts  of  the  slab  unequally. 

Taking  up  now  the  deferred  proof  that  the  diagonal  rods  where 
they  cross  the  edge  of  the  octagon  are  subject  ultimately  to  the  same 
local  stresses  and  strains  as  the  direct  rods  of  the  side  belts;  note 
that  these  diagonal  rods  lie  in  a  triangular  area  between  two  side 
belts,  which  latter  experience  equal  elongations  e\  in  directions  at 
right  angles  to  each  other.  The  edges  of  the  triangle  in  which  the 
single  layer  of  diagonal  rods  lie  are  continuous  with  the  side  belts 
and  necessarily  experience  the  same  elongations,  which  are  propo- 
gated  from  the  side  belts  into  the  triangle  by  the  agency  of  horizontal 
shears  on  its  edges.  Such  equal  elongations  at  right  angles  imply 
the  same  elongation  in  every  direction  in  the  triangle,  as  appears 
from  the  fundamental  properties  of  equal  principal  stresses  and 
strains.  Hence  we  have  the  same  elongations  along  the  diagonal 
rods  as  along  the  rods  of  the  side  belts  at  the  edges.  The  existence 
of  an  ultimate  stress  and  strain  in  the  diagonal  belt  equal  to  that  in 
the  side  belt  would  require  that  the  cross  sections  A2  and  AI  of  the 
two  belts  should  be  equal,  altho  so  far  as  the  elastic  value  of  fa  at 
the  center  of  the  panel  is  concerned  A2  might  be  less  than  AI,  as  has 
been  already  shown  in  (52)  and  (53).  The  relationships  of  stress, 
load,  etc.,  for  this  ultimate  condition,  have  been  already  given  in  (37). 

Besides  the  stresses  and  strains  in  the  diagonal  belts,  just  in- 
vestigated, those  due  to  the  local  stretching  (arising  from  the  deflec- 
tions themselves)  exert  their  greatest  effect  on  the  rods  of  the  diagonal 
and  side  belts  just  in  the  region  of  the  line  of  weakest  section,  and 
partly  because  of  that  fact.  While  these  local  stresses  may  not  exceed 
10%  in  addition  to  those  already  present,  their  existence  should 
prevent  any  thought  of  taking  2 A  larger  than  AI  in  (37)  when  deriv- 
ing the  ultimate  stresses  at  the  yield  point.  Similar  results  may  be 
formulated  to  cover  cases  where  g  is  greater  or  less  than  7/16  L. 

It  is  perhaps  desirable  at  this  point  to  consider  a  little  more  at 
length  the  matter  of  local  stretching  in  a  slab.  It  is  impossible  for  a 
continuous  flat  floor  slab  to  undergo  the  deflections  which  we  are 
treating,  consisting  of  convexities,  concavities,  etc.,  without  local 
stretching  to  allow  this  to  occur.  A  floor  slab  of  many  panels  does 
not  undergo  any  change  of  its  total  linear  dimensions  which  would 
account  for  these  corrugations.  A  continuous  beam  under  flexure 
would  have  its  extremities  drawn  toward  each  other.  But  not  so 


ACTUAL   DEFLECTIONS    IN    SIDE    BELTS  199 

to  any  such  extent  with  a  slab.  Such  contractions  are  resisted  by 
local  circumferential  strains  which  result  in  true  stresses.  An 
investigation  of  such  stresses  leads  to  the  conclusion  just  stated  that 
in  general  they  cannot  exceed  10%  of  the  ordinary  stresses  due  to 
slab  bending  when  they  are  left  out  of  the  consideration.  For  this 
reason  a  single  panel  alone  will  not  function  precisely  in  the  same 
way  as  a  panel  in  a  floor  of  many  panels. 

12.  Actual  deflections  are  distances  which  any  given  points 
of  a  slab  sink  down  by  reason  of  the  application  of  a  given  load,  and 
their  theoretical  values  are  to  be  computed  by  help  of  the  formulas 
which  have  been  developed  for  z  in  the  various  areas  into  which  the 
panel  has  been  divided. 

We  shall  now  make  a  slight  modification  in  our  definition  of  the 
level  of  the  origin  of  coordinates,  and  shall  take  it  at  the  upper  or 
lower  plane  surface  of  the  flat  slab  before  flexure,  in  which  surface 
the  axes  of  x  and  y  are  assumed  to  lie.  It  is  of  no  consequence 
whether  it  be  the  upper  or  the  lower  surface  which  is  assumed,  the 
equations  will  be  the  same  in  either  case.  The  reason  for  this  new 
definition  of  the  position  of  the  origin  is  this:  Each  kind  of  partial 
area  into  which  the  slab  has  been  supposed  to  be  subdivided  has  its 
neutral  surface  at  a  different  depth  in  the  slab,  and  so  it  does  not 
furnish  a  single  suitable  level  from  which  to  reckon  deflections,  as 
does  the  upper  or  lower  surface  of  the  slab.  None  of  the  equations 
which  have  been  derived  in  this  paper  will  undergo  any  modification 
by  reason  of  this  change  of  definition.  It  has  been  assumed  that 
each  kind  of  area  has  a  separate  value  of  /  which  remains  constant 
thruout,  so  that  the  neutral  surfaces  of  different  areas  do  not  join 
at  their  edges.  As  previously  explained  this  is  of  no  consequence 
mechanically  by  reason  of  the  zero  true  moments  that  exist  at  these 
edges.  The  modification  just  introduced  avoids  the  geometrical 
perplexities  arising  from  this  discontinuity  of  neutral  surfaces. 

Deflections  in  the  side  belt  area  between  the  lines  of  contra- 
flexure  (24)  are  to  be  found  from  (30),  or  (31),  and  (32).  To  find 
the  deflection  or  difference  of  level  in  the  mid  side  belt  between 
x  =  0,  y  =  b,  and  x  =  |  a  V~3,  y  =  b,  substitute  these  values  in  (30), 
take  i  =  0.71,  j  =  0.91,  K  =  0.5  and  subtract  the  value  z  at  the 
second  point  from  that  at  the  first  point,  which  gives  the  following 
value  of  the  deflection  of  the  one  point  below  the  other: 

W  L\ 


200  ACTUAL   DEFLECTIONS    IN    CENTRAL   AREA,    IN    HEAD 

in  which  di  is  the  vertical  distance  from  the  center  of  the  single  belt 
of  rods  at  the  mid  span  of  the  side  belt  to  the  effective  top  of  the 
slab,  considering  the  strip  fill  or  other  concrete  finish  at  its  effective 
value. 

In  the  same  manner  take  the  difference  of  level  in  the  central 

rectangle  bounded  by  the  lines  of  contraflexure  between  the  center 

point  at  x  =  0,  y  =  0  and  the  corner  x  =  I  a  Vs,  y  =  \  b  V^3  by  using  (21) 

and  (51)  and  introducing  the  values  i  =  2/3,     j  =  0.89,  etc.,  and 

C2  =  1/4(L!/L2  +  !)(!+  La /Li),  then: 

C2W  L\ 
A*  =  6.56x10'°  eg  A, ' (55) 

in  which  A2  is  the  cross  section  of  one  diagonal  belt  and  d2  is  the 
vertical  distance  from  the  center  of  the  upper  or  second  diagonal 
belt  to  the  effective  upper  surface  of  the  panel  at  its  center. 

On  evaluating  C2  above,  we  find 
when        l>L2/Li  >0.75 
then         1>      C2       >0.77 

hence  we  may  with  sufficient  accuracy  for  practical  purposes  assume 
C2  =  La/Li (56) 

Deflections  in  the  mushroom  area  between  lines  of  contraflexure 
(24)  are  to  be  derived  from  (39)  (40)  and  (41)  by  introducing  i  =  i, 
j  =  0.83,  K  =  0.5  and  1<A  =  7.5  AI.  Assuming  the  diameter  of 
the  cap  to  be  0.2Li  we  have,  at  its  edge  where  x  =  O.Sa,  y  =  b,  from 
(39) 

W  L\  (Li/La  +  1)  /36 

19.1  x  1010  "(57) 


)/36y 
x \ioo/ 


The  value  of  z  at  the  edge  of  the  mushroom  area,  where  x  =  §  a  V%  , 
y  =  b,  is  to  be  obtained  from  (57)  by  replacing  the  last  factor  by 
4/9;  and  the  deflection  between  the  edge  of  the  cap  and  the  edge  of 
the  mushroom  obtained  by  taking  the  difference  of  these  quantities 
is  as  follows: 


(58) 


in  which  h<3  is  the  vertical  distance  of  the  center  of  the  third  layer  of 
reinforcing  rods  over  the  edge  of  the  cap  above  the  bottom  surface 
of  the  slab. 


TOTAL  DEFLECTIONS  BELOW  EDGE  OF  CAP  201 

Similar  expressions  may  be  obtained  for  the  values  of  z  and  Az 
on  the  side  parallel  to  y,  where  x  =  a  at  y  =  0.86,  and  y  =  J  6^3, 
by  exchanging  LI  and  L2  in  (57)  and  (58). 

Take  half  the  sum  of  (57)  and  the  corresponding  values  so  ob- 
tained at  x  =  a,  y  =  0.86,  as  the  value  of  z  at  the  edge  of  the  cap 
where  it  is  intersected  by  the  diagonal  of  the  panel,  viz. 

TF  (L^  +  L2)  (L\  +  L\)  /  36  y 
"  38.2  x  1010  L!  L2d\Al  \1007 

and  subtract  this  from  the  value  of  z  on  the  diagonal  at  the  corner 
of  the  mushroom  area  where  x  =  \  a*/3,  y  =  i  6V^3  and  we  have 

C2W  L\ 

A*-I£7w=^--. (60) 

as  the  deflection  along  the  diagonal  between  the  edge  of  the  cap  and 
the  intersection  of  the  lines  of  contraflexure,  in  which  C2  and  ^3  are 
as  previously  defined. 

Expressions  (58)  and  (60)  somewhat  exceed  the  true  values  of 
these  deflections  because  the  slab  has  no  slope  at  the  edges  of  the 
cap  as  tacitly  assumed.  A  close  estimate  requires  that  the  denom- 
inators be  increased  by  60  percent  on  this  account,  thereby  chang- 
ing the  factors  60  to  96  and  12.5  to  20  respectively.  These  amended 
values  of  the  deflections  will  be  used  hereafter  instead  of  (58)  and  (60) . 

Let      Dl  = 
and      D2  = 

in  which  D1  is  the  deflection  of  the  mid  point  of  the  side  belt  below 
the  edge  of  the  cap,  and  D2  is  the  deflection  of  center  of  the  panel 
below  the  edge  of  the  cap. 

The  proportionate  deflections  of  these  points  are  obtained  by 
dividing  by  the  spans,  viz:  Dl/Li  and  D2 /  *L\  +  1%. 

13.  Estimated  proportionate  deflections  may  be  obtained  from 
(61)  under  such  circumstances  as  to  convey  reliable  information 
respecting  what  may  be  reasonably  expected.  Let  h  =  the  total 
thickness  of  the  slab.  The  limiting  values  of  the  thickness  of 
standard  mushroom  construction  are  expressed  as  follows: 

L!  /20>h>L1  /35, (62) 

and  assuming  that  the  reinforcing  rods  are  1/2"  rounds  with  1/2" 
covering  of  concrete  we  shall  have  from  the  definitions  of  di,  d2  and 
d3)  already  given 


202 


PROPORTIONATE  DEFLECTIONS 


h  =  h  +  0.75  =  d2  +  1.25  =  d3  +  1.75  ..........  (63) 

Substituting  these  in  (62)  etc.  we  have 

Li/20  —  0.75  >  d1  >  Li/35  —  0.75   ] 

Li/20  —  1.25  >  d2  >  Li/35  —  1.25    }>  .........  (64) 

Li/20  —  1.75  >  d3>  Li/35  —  1.75   J 

If  it  be  assumed  that  we  are  dealing  with  medium  sized  panels 
about  20'  x  20'  (64)  ,  may  be  written  in  the  form  : 

(1  —  0.062)  Li/20  >  di  >  (1  —  0.11)  Lx/35 
(1  —0.1)  Li/20    >d2  >  (1  —0.18)  Li/35 
(1  —  0.15)  Li/20  >  d3  >  (1  —  0.255)  Li/35 


0.94      di        0.89 
or,  ->-->- 


20        LI          35 

0.90      d2        0.82 

-  >  —  >  -       -• 
20        L  35 


0.85      d3        0.745 
_  \  _  \  __ 

20        Li         35 


(65) 


In  (54),  (55),  (58)  and  (60)  replace  W 
viz,    175  di  AI  /„,  and  we  have 

A  2,  = 


A  z2 
A  z3 


6.11  x  108  dfi 
C2  di  LI  Aifs 


3.75  x  108  d\  A 


2 


dl 


by  its  value  given  in  (34), 


55  x  108  d 

n    j    72  f 
C2  «i  LI  js 

11.4x  108fl 


in  which  /8  is  the  greatest  stress  in  the  steel,  i.  e.,  at  the  mid  side 
belt,  employed  here  to  express  deflections  instead  of  expressing  them 
in  terms  of  panel  load  as  was  done  previously. 


. . (66) 


RELATIVE    DEFLECTIONS    AT   MID    SPAN 


203 


Introduce  into  (66)  the  numerical  values  given  in  (65)  which 
will  then  express  limiting  values  of  deflection  for  medium  spans. 
For  simplicity  let  LI  =  L2  then: 


287   > 


>    155 


162   >  >    81 

105Az2 


1056   > 


>    498 


Llfa 
440   >  >    203 

105Az4 


(67) 


By  (61)  we  have  the  proportionate  deflection  of  the  side  and  diagonal 
belts  as  follows: — 


r  i      i  i  /.     0i    r  i      11 

+  + 

1.287       1056J  10s        LI       1.155      498J 


f* 


[1        1 
162       44C 

1- 

J  105  V< 

£>2 

I      Li  V2      L^ 

/s 

£ 

L            1  I      /s 

h 
1         203J  105  V2 

225  x  105 

118  x   105 

/. 

/. 

167.  4  x  105 
Kf          i  Annn 

LVO 
1     ^ 

1 

82  x    105 
D2             1 

Js    —    IbUUU, 
If  /s   =   24000 

1046 
1 

LI  V2     '   512 
D2             1 

Kf          Qonnn 

697 
1 

LI  V2        341 
D2             1 

Js    --    6Z(J(J(J, 

523 

L!  V2     "   256 

(68) 


(69) 


Larger  spans  then  20 ',  or  smaller  steel  than  1/2"  round,  or  L2<L>i 
will  reduce  the   above   values  somewhat,  while  smaller  spans   or 


204  THEORETICAL    AND    EXPERIMENTAL   DEFLECTIONS    COMPARED 

larger  steel  will  increase  these  values,  all  of  which  can  in  each  case 
be  submitted  to  calculation  by  the  methods  here  developed. 

To  recur  at  this  point  to  the  expression  for  the  deflection  D2  in 
terms  of  the  panel  load  W  by  help  of  (55),  (60)  and  (61) 

C2  W  L\ 

(70) 


11 

dl  J 


By  (65)  we  find 


90       d2         82  dz 

-<        -  ,   or  1 . 1  >  -  -   >  1 .  06 
85       d3        74.5  d3 

and  using  this  inequality  to  eliminate  d^  from  (70)  we  find  after 
reduction 

C2  W  L\  C2W  L\ 

4.8  x  1010  d22  A,  <  4.7  x  1010  d\  A, 

from  which  we  may  write  as  a  mean  value 

C2  W  L\ 
D2=  (71) 


The  empirical  deflection  formula  given  on  page  29  of  Turner's  Con- 
crete Steel  Construction,  when  written  in  these  units,  is 

WL\ 


This  is  identical  with  (71)  when  C2  =  0.98,  and  diverges  from  it 
slightly  for  other  admissible  values  of  C2.  The  practical  agreement 
of  (71)  and  (72)  affords  a  second  confirmation  of  the  theoretical 
deductions  made  thus  far,  and  this  taken  in  conjunction  with  the 
practical  identity  of  formulas  (34)  and  (35),  the  theoretical  and 
empirical  expressions  for  the  maximum  tensile  stresses  in  the  rein- 
forcement, furnishes  what  on  the  theory  of  probabilities  may  be 
regarded  as  so  strong  a  probability  of  the  general  trustworthiness 
of  the  entire  theory  as  to  exclude  any  rational  supposition  to  the 
contrary. 

The  various  formulas  for  stresses  and  for  deflections  which  have 
been  developed  in  this  paper  have  been  obtained  under  the  express 
proviso  that  the  panel  under  consideration  was  assumed  to  be  one 
of  a  practically  unlimited  number  of  equal  panels  constituting  a 
continuous  slab,  all  of  which  are  loaded  uniformly  and  equally.  The 


MUSHROOM    PANELS    INDEPENDENT  205 

question  at  once  arises  as  to  the  amount  and  kind  of  deviations  from 
these  formulas  which  will  occur  by  reason  either  of  discontinuity  of 
slab  or  loading,  such  as  occurs  at  the  outside  panels  of  a  slab  or  at 
panels  surrounded  partly  or  entirely  by  others  not  loaded.  The 
answer  to  this  question  depends  very  largely  upon  the  construction 
of  the  flat  slab  itself. 

In  the  standard  mushroom  construction  it  has  been  found  that 
the  stresses  and  deflections  of  any  panel  are  almost  entirely  inde- 
pendent of  those  in  surrounding  panels.  This  is  due  to  the  fact 
that  the  mushroom  head  is  an  integral  part  of  the  supporting  column 
in  such  a  manner  that  it  is  impossible  for  it  to  tilt  appreciably  over 
the  column  under  the  action  of  any  eccentric  or  unequal  loading  of 
panels  near  it.  When  single  panels  have  been  loaded  with  test 
loads,  no  appreciable  deflections  have  been  discoverable  in  sur- 
rounding panels,  and  no  greater  stresses  and  deflections  have  been 
discovered  than  were  to  be  expected  in  case  surrounding  panels  were 
loaded  also.  Future  careful  investigation  of  this  may  reveal 
measureable  effects  of  this  kind,  but  they  must  be  small. 

A  like  statement  cannot  be  made  of  other  systems  of  flat  slab 
construction  where  the  reinforcement  over  the  top  of  the  column  is 
not  an  integral  part  of  the  column  reinforcement  itself.  Tests  on 
these  systems  have  shown  clearly  the  effects  of  the  tipping  of  the 
part  of  the  slab  on  the  top  of  the  column,  and  lack  of  stiffness  of 
head,  in  the  increase  of  the  deflection  of  the  single  loaded  panel  over 
the  deflection  to  be  expected  in  case  of  multiple  loaded  panels,  and 
especially  in  the  disturbance  of  the  equality  of  the  stress  in  the  other- 
wise equal  stresses  in  the  rods  of  the  side  belts.  Such  distrubance, 
by  increasing  the  stress  in  part  of  these  rods,  would  necessitate  larger 
reinforcement  in  the  side  belts  of  such  systems  than  would  be  re- 
quired in  mushroom  slabs.  The  great  stiffness  of  the  mushroom 
head  is  also  of  prime  importance  in  taking  care  of  accidental  and 
unusual  strains  liable  to  occur  in  the  removal  of  forms  from  under 
insufficiently  cured  slabs. 

14.  In  considering  the  design  of  the  ring  rods  and  radial 
cantilever  rods  of  the  mushroom  head,  it  should  be  borne  in  mind 
that  they  occupy  a  position  in  such  close  proximity  to  the  level 
of  the  neutral  surface  as  to  prevent  them  from  being  subjected 
to  severe  tensile  or  compressive  stresses  by  reason  of  the  bending 
of  the  slab  as  a  whole.  Their  principal  function  as  slab  mem- 
bers is  to  resist  shearing  stresses  and  the  bending  stresses  due  to 


206  VERTICAL  SHEAR  AROUND  COLUMN  CAPS 

local  bending.  Their  total  longitudinal  stresses  are  too  small  in 
comparison  to  require  consideration. 

Let  a  cylindrical  surface  be  imagined  to  be  drawn  concentric 
with  a  column  to  intersect  the  slab,  then  the  total  vertical  shearing 
stress  which  is  distributed  on  the  surface  of  intersection  is  equal  to 
the  total  panel  load  W  diminished  by  the  amount  of  that  part  of  the 
panel  load  lying  inside  the  cylinder.  If  the  cylinder  be  not  large, 
the  total  shear  may  be  taken  as  approximately  equal  to  W  itself. 

It  is  evident  that  the  smaller  the  diameter  may  be  that  is 
assumed  for  this  cylinder,  the  greater  will  be  the  intensity  of  the 
vertical  shear  on  its  surface  and  that  for  two  reasons :  First,  because 
the  tot?l  load  thus  carried  to  the  column  will  be  greater  the  smaller 
the  diameter,  and  second  because  the  surface  over  which  the  total 
shear  will  be  distributed  decreases  with  its  diameter. 

The  result  of  this  is  that  the  dangerous  section  for  shear  is  the 
cylindrical  surface  at  the  edge  of  the  cap.  For  cylinders  smaller 
than  this  the  increased  vertical  thickness  of  the  cap  diminishes  the 
intensity  of  the  shear.  We  proceed  therefore  to  consider  the  manner 
in  which  the  total  vertical  shearing  stress  of  approximately  W  in 
amount  is  distributed  in  the  material  of  the  cylindrical  surface  at  the 
edge  of  the  cap. 

In  a  beam  or  slab  the  horizontal  shearing  stresses  due  to  bending 
reach  a  maximum  at  the  neutral  surface.  It  is  a  fundamental  con- 
dition of  equilibrium  that  shearing  stresses  on  planes  at  right  angles 
shall  be  equal,  and  it  is  this  condition  that  determines  the  distribu- 
tion of  the  vertical  shears,  which  are  at  right  angles  to  the  horizontal 
shears  resulting  from  bending  the  slab  as  a  whole.  From  this  we 
have  the  well  known  fact  that  the  vertical  shear  varies  from  zero 
at  the  upper  and  lower  surfaces  to  a  maximum  at  the  neutral  surface, 
and  this  is  necessarily  the  manner  in  which  the  total  shear  is  dis- 
tributed at  the  edge  of  the  cap.  The  top  belt  of  rods  will  be  sub- 
jected to  comparatively  small  shearing  stresses,  and  successive 
layers  of  rods  will  be  under  larger  and  larger  shearing  stresses  by 
reason  of  their  greater  nearness  to  the  neutral  surface,  while  the 
total  shear  borne  by  the  radial  rods  near  the  neutral  surface  will  be 
much  larger  than  that  upon  the  others.  The  shearing  stress  in 
the  concrete  will  need  to  be  considered  also. 

It  is  to  be  noticed  that  all  the  steel  of  the  belts  and  mushroom 
head  act  together  without  the  necessity  of  supposing  large  com- 
pressive  stresses  in  the  concrete  to  transmit  vertical  forces,  because 
the  belts  of  reinforcement  rest  directly  upon  each  other,  and  these 
in  turn  upon  the  ring  rods  and  radial  rods,  all  in  metallic  contact 


VERTICAL  SHEAR  IN  RADIAL  RODS,  ETC.  207 

with  each  other,  in  the  mushroom  head,  and  so  they  transmit  and 
adjust  the  distribution  of  stresses  within  the  system  to  a  very  large 
extent  independently  of  the  concrete. 

We  can  then  safely  assign  moderate  values  of  the  shearing 
stress  to  each  of  the  elements  that  constitute  the  slab  at  the  edge 
of  the  cap,  with  the  assurance  that  they  will  each  play  a  part  in 
general  accordance  with  the  distribution  which  has  been  already 
explained. 

The  mushroom  is  constructed  of  great  strength  and  stiffness 
not  merely  to  effect  the  results  which  have  appeared  previously  in 
the  course  of  the  investigation  but  also  to  ensure  the  stability  of  the 
slab  in  case  of  unexpected  or  accidental  stresses  due  to  the  too  early 
removal  of  the  forms,  before  the  slab  is  well  cured,  at  a  time  when 
the  only  load  to  which  it  is  subjected,  is  due  to  the  weight  of  the 
structure  itself. 

The  working  load  to  be  assumed  in  designing  the  mushroom 
may  be  taken  as  the  dead  load  of  a  single  slab  plus  the  design  load, 
provided  sufficiently  low  values  of  the  shearing  stresses  be  assumed 
in  the  cross  sections  of  steel  and  concrete  at  the  edge  of  the  cap 
for  the  support  of  this  working  load,  as  follows : 

For  slabs  having  a  thickness  of  h  =  L/35  a  mean  working 
shearing  stress  of  2000  Ibs.  per  square  inch  at  the  right  cross  section 
of  each  reinforcing  rod  which  crosses  the  edge  of  the  cap,  a  mean 
shearing  stress  of  40  Ibs.  per  square  inch  in  the  vertical  cylindrical 
section  of  the  concrete  at  the  edge  of  the  cap,  and  8000  Ibs.  per 
square  inch  of  right  cross  section  of  each  radial  rod. 

For  slabs  having  a  thickness  of  h  =  L/20  the  intensities  just 
given  may  be  safely  increased  by  50  per  cent  for  reasons  that  will 
be  explained  later.  For  slabs  of  intermediate  thickness  increase 
the  intensities  proportionately. 

These  values  are  sufficiently  low  to  enable  the  structure  to  sup- 
port itself  before  the  concrete  is  very  thoroughly  cured,  and  the 
head  so  designed  will  be  found  after  it  is  well  cured  to  be  so  pro- 
portioned as  to  carry  safely  a  test  load  of  double  the  live  and  dead 
loads  for  which  it  was  designed. 

In  this  connection  it  seems  desirable  to  investigate  what  takes 
place  in  case  of  overloading  and  incipient  failure  of  an  insufficiently 
cured  slab,  or  one  unduly  weakened  by  thawing  of  partially  frozen 
concrete.  Suppose  that  under  such  circumstances  a  shearing  crack 
were  formed  extending  completely  thru  the  head  at  the  edge  of  the 
cap,  and  we  wish  to  investigate  the  stresses  and  behavior  of  the  rods 


208  STRESSES    IN    RADIAL   RODS 

that  cross  the  crack  at  which  shearing  deformation  has  begun  to  take 
place.  Designate  the  position  of  the  crack  by  X. 

The  total  vertical  shearing  stress  on  a  radial  rod  at  X  is  the 
sum  of  two  parts  found  as  follows:  First,  the  vertical  reaction  at 
the  top  of  a  column  is  made  up  of  the  vertical  reaction  of  the  con- 
crete core  of  the  column  and  the  reactions  of  its  vertical  reinforcing 
rods.  Call  the  vertical  reaction  of  one  of  these  rods  Vi.  The  rod 
is  bent  over  radially  and  Vi  expresses  also  the  amount  of  the  vertical 
shear  in  that  rod  where  it  starts  out  radially  from  the  column. 
Between  this  point  and  X  for  a  distance  which  measures  usually 
from  9  to  12  inches,  the  rod  experiences  the  supporting  pressure  of 
the  concrete  in  the  cap  under  it  to  a  total  amount  which  we  will 
designate  by  V2.  The  total  shear  in  the  radial  rod  at  X  will  then 
amount  to 

V    -  V,  +  Fa..  (73) 

provided  we  neglect  the  weight  of  that  small  part  of  the  actual  load 
of  the  slab  which  lies  directly  over  this  piece  of  the  rod  and  may 
be  regarded  as  resting  upon  it.  This  portion  of  the  radial  rod  of 
length  I  is  a  cantilever  fixed  at  one  end  in  the  top  of  the  column,  and 
carrying  a  load  V  at  the  other  end  with  a  supporting  pressure  under- 
neath of  total  amount  V2  whose  intensity  is  greatest  at  X  and  gradu- 
ually  decreases  along  I  from  X  to  the  fixed  end.  The  rod  has  a 
point  of  contraflexure  and  zero  moment  at  X.  The  portion  of  the 
rod  outside  the  crack  has  a  fixed  point  in  the  slab  at  the  place  where 
it  supports  the  inner  ring  rod,  at  a  distance  from  X  which  should 
not  exceed  I  as  just  defined.  Similar  conditions  hold  for  this  length; 
i.  e.  there  will  be  a  total  shear  in  the  radial  rod  at  a  point  just  inside 
the  inner  ring,  rod  due  to  its  total  shear  outside  this  ring  rod  and  to 
the  vertical  loading  imparted  to  it  by  the  ring  rod  itself.  To  this 
must  be  added  the  downward  pressure  of  the  concrete  between  the 
inner  ring  rod  and  X.  All  these,  together,  constitute  the  total 
shear  — F  at  X,  in  equilibrium  with  the  reaction  +  V  already  ob- 
tained at  that  point. 

We  shall  discuss  separately  the  action  of  V\  and  F2  upon  a  radial 
rod.  A  load  V\  at  the  end  of  a  cantilever  of  length  I  causes  a 

deflection  of  amount  z1  =  J  Vl  f  /El (74) 

in  which  1=  TT  £4/64  where  Z  =  the  thickness  of  the  rod. 

Also  FI  =  SI  A     ,  A=  7r£2/4 

in  which  Si  =  the  mean  shearing  stress  per  square  unit  of  cross  sec- 
tion and  A  is  the  cross  section  of  the  rod.  Hence 

si  =  3«i  E  12/1Q? (75) 


STRESSES    IN    RADIAL   RODS  209 

which  shows  that  so  far  as  V\  is  concerned,  for  any  given  displace- 
ment Zi  the  shearing  stress  carried  per  square  unit  of  rod  will  be  pro- 
portional to  the  square  of  its  diameter,  and  up  to  its  permissible 
limiting  shearing  resistance,  each  unit  of  section  of  such  a  rod  will 
be  effective  in  proportion  to  the  square  of  its  diameter.  For  econ- 
omical construction,  this  will  require  the  radial  rods  to  be  few  and 
large,  rather  than  numerous  and  small.  The  bending  moment  is 
greatest  at  the  distance  I  from  X  and  amounts  to  V\  I.  The  stress 
in  the  extreme  fiber  due  to  the  bending  moment  V\  I  in  the  rod  is 
Pi  =--  Vi  I  t/2I  =  8«i  l/t  .......................  (76) 

This  equation  shows  that  the  stress  in  the  extreme  fiber  is  so  very 
large  at  the  fixed  end  of  the  rod  compared  with  the  shear  at  X  that 
so  far  as  V\  is  concerned  the  rod  will  suffer  permanent  deformation 
by  bending  long  before  there  is  any  danger  of  its  shearing.  V\  is  so 
large  compared  with  V2  that  this  conclusion  will  not  be  altered  when 
we  come  to  consider  the  combined  action  of  V2. 

Incipient  failure  of  this  kind  will  therefore  cause  distortion  and 
sag  without  collapse.  In  case  such  sag  as  occurs  in  this  case  is 
detected  underneath  the  head  around  the  cap,  the  slab  should  be 
blocked  up  at  once  and  the  concrete  picked  out  at  all  parts  showing 
facture.  This  should  then  be  refilled  with  a  stronger  concrete 
which  will  set  rapidly.  Such  repair  should  not  weaken  the  slab. 

Whenever  the  intensity  with  which  a  radial  rod  presses  upon 
the  concrete  at  the  edge  of  a  crack  at  X  passes  the  compressive 
strength  /„  of  the  concrete,  it  must  begin  to  yield.  At  this  instant 
we  shall  have  a  pressure  of  the  concrete  against  the  rod  which  gradu- 
ally diminishes  as  we  pass  along  the  rod  from  X  to  the  distance  I, 
where  it  becomes  zero.  We  shall  assume  that  the  pressure  dimin- 
ishes uniformly  with  this  distance.  This  may  not  be  precisely  cor- 
rect, but  cannot  be  much  in  error.  If  the  shear  V2  at  X  is  the  sole 
cause  of  this  pressure,  then  V2  =  ^tlfc,  and  f  V2  I  =  %  t  I2  fc 
is  the  bending  moment  in  the  rod  at  the  distance  I,  due  to  V2  at  X 
and  the  pressure  distributed  along  L 

It  will  be  found  that  these  produce  a  deflection 

z2  =  3  /c  14/2QEI  =  0.3  /3  V2/EI  ................  (77) 


a  unit  shear  of 

S2  =  V2/A  =  z2E  *2/4.8  I3  ....................  (78) 

and  a  stress  on  the  extreme  fiber  at  a  distance  I  amounting  to 

p2  =  V2  1  1/31  =  16s2  l/t.  .....................  (79) 


210  SHEAR    IN    OUTER   RING    RODS 

It  thus  appears  that  the  equations  expressing  the  action  of  F2 
are  precisely  similar  to  those  for  V\,  differing  only  in  their  numerical 
coefficients,  and  consequently  all  the  statements  as  to  the  resistance 
of  the  radial  rods  under  the  action  of  Vi  hold  for  the  action  of 
Vi  and  V2  together  in  the  case  of  given  initial  deformations, 
Zi  =  Z2  at  X. 

While  the  preceding  investigation  has,  in  order  to  make  ideas 
explicit,  ostensibly  assumed  a  crack  at  X  and  an  initial  small  shear- 
ing deformation  at  X,  the  investigation  applies  equally  well  to  the 
elastic  shearing  deformation  of  the  concrete  at  the  dangerous 
section  in  which  case  the  total  shearing  stress  will  consist  of  an  addi- 
tional componenent  due  to  the  resistance  of  the  concrete,  which 
however  may  for  additional  safety  be  neglected.  If  the  assumed 
deformation  be  confined  within  limits  so  small  that  the  concrete 
is  able  to  endure  it  without  cracking  then  the  preceding  investiga- 
tion may  properly  be  applied  to  it.  It  is  right  here  that  the  thick- 
ness of  the  radial  rods  is  able  to  render  its  most  effective  service, 
for  it  appears  from  (75)  and  (78)  that  any  permissible  intensity  of 
shear  may  be  developed  in  the  radial  rods  by  making  them  of  suit- 
able thickness,  even  tho  the  deflection  be  kept  within  the  elastic 
limits  of  the  concrete. 

As  already  stated  we  must  not  overlook  the  fact  that  the  major 
stresses  here  are  those  under  the  head  of  Fj,  which  are  due  to  the 
direct  metallic  contacts  of  the  steel  rods  resting  one  upon 
another,  where  large  stresses  are  transmitted  and  pass  independ- 
ently of  the  concrete  except  for  the  distortions  of  the  steel  which 
meet  resistance,  and  the  secondary  reactions  such  as  have  been 
treated  in  a  single  aspect  while  investigating  the  action  of  V2. 

It  is  due  to  this  fact  that  large  shearing  stresses  may  be  safely 
borne  by  the  slab  at  and  near  the  edge  of  the  cap,  which  the  concrete 
mostly  escapes,  it  merely  furnishing  some  lateral  stiffening  to  the 
steel.  On  this  principle  the  outer  ring  rod  should  have  a  cross 
section  not  much  less  than  one  half  that  of  the  radial  rods  on  which 
it  rests.  For,  this  arrangement  provides  for  the  transferal  to  the 
radial  rods  of  all  the  shear  the  ring  rod  is  able  to  carry,  it  being  in 
double  shear  compared  with  the  radial  rod  it  rests  on. 

It  is  impossible  to  determine  the  cross  section  of  the  inner  ring 
rod,  with  the  same  definiteness  as  that  of  the  radial  rods,  but 
that  is  unimportant.  Its  position  has  already  been  fixed  as  not 
more  than  I  from  the  edge  of  the  cap,  where  Z  is  the  distance  from 
the  top  hoop  or  collar  band  of  the  column  to  the  edge  of  the  cap. 


STRESSES    IN    CONCRETE    OF    HEAD  211 

The  vertical  shearing  stresses  may  be  regarded  as  sufficiently 
resisted  outside  the  mushroom  by  the  concrete  alone.  The  critical 
cylindrical  surface  separating  those  areas  where  the  shear  may  be 
assumed  to  be  safely  carried  by  concrete  alone,  from  those  areas 
where  the  steel  may  be  relied  on  to  carry  as  much  of  the  shear  as 
may  be  required,  should  evidently  be  taken  somewhat  inside  the 
outer  ring  rod,  but  just  where  is  of  no  particular  consequence. 

The  supposition  of  the  existence  of  a  crack  at  X,  either  actual 
or  potential,  on  which  our  computation  of  the  stresses  in  the  radial 
rods  has  been  based,  is  sufficiently  satisfactory  so  far  as  the  rods 
themselves  are  concerned;  but  it  seems  desirable  to  consider  in  more 
detail  the  phenomena  attending  the  development  of  the  stresses  in 
the  concrete  at  and  near  the  edge  of  the  cap,  especially  in  soft  con- 
crete when  the  limit  of  its  compressive  resistance  is  reached  in  this 
region. 

The  horizontal  compressive  resistance  of  the  concrete  at  the 
lower  surface  of  the  slab  is  that  already  treated  in  (38),  and  it  is  our 
present  object  to  consider  how  that  is  to  be  combined  with  the 
vertical  supporting  pressures  under  the  radial  rods,  and  with  the 
horizontal  and  vertical  shears  in  the  slab  due  to  bending.  These 
latter  are  greatest  in  the  neutral  surface,  as  has  been  previously 
stated,  and  according  the  general  theory  of  stresses  are  equivalent 
to,  and  may  be  replaced  by,  a  compression  and  a  tension  in  the  mate- 
rial respectively  at  45°  with  the  vertical  (and  mutually  at  right 
angles)  of  the  same  intensity  as  the  shear.  It  is  evident  that  the 
combination  and  resultant  of  these  three  compressive  stresses 
would  form  the  dangerous  element  in  the  stress,  since  the  single 
tensile  element  would  be  relatively  unimportant,  and  it  would  find 
assistance  in  its  resistance  from  the  steel  running  in  a  direction  thru 
the  concrete  such  as  to  afford  it  substantial  support.  This  direction 
is  that  of  the  straight  lines  on  the  surface  of  a  right  cone  whose 
vertex  is  above  the  center  of  the  column  and  whose  slope  is  1  to  1 . 

Consider  now  two  of  the  elements  of  the  compression  in  the 
concrete  around  the  cap,  viz,  the  horizontal  compression  which  is  a 
maximum  at  the  lower  surface  and  zero  at  the  neutral  surface,  and 
that  due  to  shear  which  is  parallel  to  the  sides  of  a  right  cone  with 
vertex  downward,  whose  sides  have  an  upward  and  outward  slope 
of  1  to  1,  while  its  intensity  is  so  distributed  that  it  is  zero  at  the 
bottom  of  the  slab  and  greatest  at  the  neutral  surface.  It  appears 
consequently  that  the  lines  of  greatest  compression  in  the  concrete 
due  to  the  combination  of  these  two  elements  of  compression  would 


212  COMPRESSION  ON  CONCRETE  OF  HEAD 

lie  in  vertical  planes  on  a  bowl  or  saucer-shaped  surface  that  is  hori- 
zontal at  the  edge  of  the  cap  and  inclined  at  a  slope  of  45°  at  the 
neutral  surface ;  and  if  the  concrete  were  to  crush  under  these  stresses 
alone,  the  surface  of  fracture  would  have  the  shape  indicated  in- 
stead of  that  of  the  cylindrical  surface  previously  assumed.  This 
change  would  not,  however,  materially  affect  the  computations  we 
have  made  of  stresses  in  steel;  it  merely  serves  to  fix  more  definitely 
the  position  of  the  points  of  contra-flexure  of  the  radial  rods. 

But  there  is  still  one  further  element  or  component  of  the  total 
compression  in  the  concrete  to  be  considered  and  combined  with 
those  just  treated  in  order  to  arrive  at  the  resultant  or  total  com- 
pression. This  componenent  is  that  due  to  the  concentrated  press- 
ures underneath  each  of  the  radial  rods.  These  rods  are  at  some 
distance  apart  circumferentially  and  so  do  not  exert  a  pressure  that 
is  uniformly  distributed  circumferentially.  Any  concentrated  stress, 
such  as  that  in  the  concrete  supporting  a  rod,  diffuses  itself  in  the 
material  in  such  a  manner  that  its  intensity  rapidly  diminishes  with 
the  distance  from  the  surface  of  the  rod,  in  accordance  the  same  law 
as  exists  in  case  of  centers  of  attraction.  Since  the  supporting  com- 
pression under  the  rods  is  vertical,  we  can  imagine  the  lines  of  great- 
est compression  in  the  concrete,  when  this  component  is  combined 
with  those  already  mentioned,  to  lie  in  vertical  planes  on  a  bowl  or 
saucer-shaped  surface  which  has  as  many  indentations  or  scollops 
around  its  edge  as  there  are  radial  rods,  at  which  indentations  the 
slope  of  the  sides  is  such  more  nearly  vertical  than  a  slope  of  45°. 
At  such  parts  of  the  surface  the  intensity  is  also  more  severe,  and 
especially  is  this  the  case  if  the  slab  is  thin  so  that  the  concentrated 
pressure  has  small  opportunity  to  distribute  itself  by  radiating  into 
a  considerable  body  of  material  before  it  reaches  the  bottom  of  the 
slab.  It  thus  comes  about  that  thick  slabs  are  enabled  to  carry 
safely  larger  intensities  of  shearing  stress  around  the  cap  than  can 
thin  slabs,  which  is  in  accordance  with  and  in  justification  of  the 
statements  already  made  as  to  permissible  shears  around  the  cap. 

The  resulting  surface  of  fracture  due  to  shear  and  compression 
around  the  cap  would  be  of  irregular  conical  shape  starting  from 
the  edge  of  the  cap  and  extending  thru  the  entire  thickness  of  the 
slab,  were  this  not  interfered  with  in  the  upper  part  of  the  slab  by 
the  mat  of  reinforcing  rods,  which  are  so  tenacious  as  to  tear  to 
pieces  and  fracture  the  upper  surface  to  a  considerable  distance  in 
all  directions  whenever  any  such  fracture  occurs  around  the  column. 

Nevertheless  such  fracture  as  here   described   does  not  under 


RADIAL    AND    RING    RODS    PREVENT    FRACTURE    OF    HEAD  213 

any  ordinary  circumstances  result  in  a  dangerous  collapse  of  the 
slab,  or  one  that  cannot  be  repaired  without  much  difficulty,  for,  the 
radial  rods  and  the  reinforcing  rods  will  at  most  have  suffered  some 
individual  deformation  by  bending  and  are  still  far  from  being 
broken.  This  will  become  evident  later  where  an  experimental 
attempt  to  load  a  full-sized  slab  to  failure  is  described  in  detail,  and 
full  account  of  the  results  reached  is  explained  and  illustrated. 

It  is  stated  on  good  authority  that  in  experience  with  many 
hundreds  of  buildings  constructed  on  this  system,  no  case  of  shear 
failure  or  even  of  incipient  shear  failure  or  fracture  has  occurred  in  a 
well  cured  slab  near  the  column  and  while  a  few  cases  of  incipient 
failure  have  occurred  in  floors  where  forms  were  prematurely  re- 
moved, no  injury  or  fatality  has  resulted  therefrom  to  any  person. 

It  appears  that  the  line  of  weakest  section  in  the  cured  slab  of 
the  standard  mushroom  type  is  that  discussed  previously  in  obtain- 
ing (37)  and  shown  in  Fig.  55,  page  164.  This  is  brought  out  later  by  a 
test  to  destruction  of  a  fairly  well  cured  slab.  The  line  of  weakest 
section  in  a  partly  cured  slab  is  on  the  other  hand  not  definitely  fixed, 
but  may  be  and  sometimes  is,  shearing  weakness  near  the  column  as 
has  been  discussed  and  pointed  out.  Provision  against  such  weak- 
ness or  carelessness  is  a  safeguard  which,  while  costing  a  small 
amount  in  the  matter  of  steel,  is  an  insurance  against  serious  acci- 
dent well  worth  the  investment  involved.  It  is  secured  by  making 
the  radial  and  ring  rods  sufficiently  stiff  and  strong. 

15.  This  section  will  be  devoted  to  a  consideration  of  the 
mushroom  system,  and  to  several  more  or  less  similar  flat  slab 
systems,  in  order  to  comment  on  the  modifications  in  mechanical 
action  that  are  produced  by  the  particular  modifications  of  the 
arrangement  of  the  reinforcement  in  these  systems. 

Fig.  53,  page  159  represents  the  section  of  a  standard  mushroom 
head  by  a  vertical  plane  thru  the  axis  of  the  column.  In  this  the 
elbow  rods  are  shown,  the  vertical  portions  of  which  are  embedded  for 
such  distances  as  may  be  necessary  in  the  columns  or  are  them- 
selves column  rods.  One  of  these  is  represented  separately  at  the 
right  side  of  Fig.  53.  They  are  confined  just  under  the  elbow  at 
the  top  of  the  column  by  a  steel  neck  band,  and  are  bent  over  at 
the  elbow  to  extend  radially  into  the  slab.  This  bent  over  portion 
is  formed  to  scale  as  to  length  and  slopes  in  accordance  with  the 
size  and  thickness  of  the  slab  in  which  it  is  to  be  used,  in  such  a 
way  that  when  the  ring  rods  and  four  layers  of  slab  rods  rest 
upon  it  and  are  tied  in  place,  the  top  of  the  upper  layer  will  be 


214  STANDARD   MUSHROOM   SYSTEM 

0.75  inch  below  the  top  of  the  slab  at  a  distance  of  the  thickness 
of  the  slab  outside  the  edge  of  the  cap,  and  at  the  same  time  the 
extremities  of  the  radial  rods  will  be  0.5  inch  above  the  bottom  of 
the  slab.  In  order  to  accomplish  this,  the  radial  portions  of  these 
rods  must  be  nearly  horizontal  over  the  cap,  and  have  a  suitable 
slope  outside  the  cap  as  shown  in  Fig.  53,  page  159. 

Fig.  55,  page  164,  shows  the  ground  plan  of  the  reinforcement  of 
the  mushroom  slab  when  the  panel  is  square  so  that  LI  =  L%  =  2a 
=  26.  In  Fig.  55  the  diameter  of  the  mushroom  head  is  assumed 
to  be  of  the  extreme  size  g  =  L/2,  a  size  which  would  increase  the 
cantilever  beyond  that  in  usual  practice  to  an  extent  not  adopted 
except  in  the  case  of  very  unusual  intensity  of  loading.  It  will 
be  observed  that  the  areas  where  the  reinforcement  consists  of  a 
single  belt  or  layer  are  thereby  rendered  small,  and  the  slab  action 
due  to  the  mutual  lateral  action  of  belts  which  cross  each  other 
exists  over  nearly  the  whole  slab. 

In  Fig.  54,  the  dimensions  of  the  rectangular  sides  are  so  taken 
that  Z/i/Z/2  =  0.75,  which  is  assumed  to  be  the  limiting  or  smallest 
value  of  that  ratio  for  constructional  purposes.  Further,  the 
diameter  of  the  mushroom  is  made  as  small  as  will  permit  the  rein- 
forcing belts  to  cover  the  entire  panel,  viz.  g  =  7  (a  +  6)/16.  For 
example  if  LI  —  20,  and  L2  =  15,  we  have  g  =  7.65+.  This  may 
be  considered  to  represent  standard  practice,  where  the  edges  of 
the  diagonal  belts  intersect  on  the  edges  of  the  side  belts.  This 
was  the  case  assumed  for  treatment  in  deriving  the  formulas  of  the 
preceeding  investigations.  Those  formulas  could  be  modified  to 
apply  to  larger  values  of  g,  by  taking  lines  of  contra-flexure  at  the 
edges  of  the  head  nearer  the  panel  center  than  given  by  (24),  and 
by  taking  larger  values  of  the  effective  cross  section  of  steel  than 
those  employed  in  (32),  (40)  and  (51). 

Now  it  is  evident  that  systems  similar  to  this  may  differ  from 
it  in  several  ways: — 

1st.  The  design  of  the  frame-work  at  the  top  of  the  column 
may.be  different  from  this  without  any  change  in  the  belts  of  re- 
inforcing rods.  It  is  hardly  possible  for  any  other  form  of  frame- 
work to  be  substituted  for  this  which  will  exhibit  the  same  rigiditj- 
of  connection  between  it  and  the  column  as  do  the  elbow  rods 
embedded  in  the  column  and  bent  over  radially  in  the  slab  so  as 
to  make  the  column  and  slab  integral  with  each  other  by  means 
of  this  common  reinforcement.  Any  reduction  of  the  stiffness  of 
connection  between  column  and  frame-work  of  head  results  in  in- 
creased tipping  of  the  head  under  eccentric  loading  of  the  slab. 


OTHER   SYSTEMS 


Fig.  56 

Eccentric  loading  is  any  loading  of  one  panel  differently  from 
another.  Tipping  of  the  head  increases  some  deflections  at  the 
expense  of  others,  and  increased  stresses  in  some  of  the  reinforcing 
rods  at  the  expense  of  others,  and  so  requires  some  additional 
reinforcement.  Such  a  frame-work  is  illustrated  in  Fig.  56,  which 
merely  rests  upon  the  top  of  the  column  without  the  support  of 
metallic  connection  with  the  vertical  column  rods.  It  consequently 
affords  less  resistance  to  tipping  under  eccentric  loads  than  when 
stiffened  by  such  metallic  connection. 

2nd.  The  ground  plan  of  the  reinforcing  belts  may  remain  un- 
changed but  part  only  of  the  belt  rods  may  be  carried  at  the  top 
of  the  slab  over  the  column  head,  while  the  rest  of  them  are  carried 
thru  under  the  head  at  the  bottom  of  the  slab.  This  modification 
of  design,  when  a  sufficient  number  of  rods  go  over  the  head  to 
resist  the  negative  bending  moments  there,  is  very  uneconomical 
of  steel,  because  in  the  case  where  they  all  go  over  the  head,  it  is 
the  fact  that  altho  the  mean  tension  of  the  steel  is  not  so  great  as 
at  mid  span,  nevertheless,  by  reason  of  the  overlapping  of  the  belts 
in  crossing,  the  stresses  in  the  rods  at  the  top  reach  a  value  not 
much  less  than  at  mid  span,  and  cannot  be  safely  diminished  in 
number.  It  thus  appears  that  the  rods  carried  thru  on  the  bottom 
are  largely  superfluous.  Of  these  two  mats  of  rods  at  top  and 
bottom,  one  of  them  is  necessarily  in  tension  and  the  other  in  com- 
pression. But  it  is  a  mistake  to  use  steel  to  resist  compression 
when  concrete  can  be  better  used  for  this  purpose.  The  lower  mat 
is  superfluous  for  this  reason. 


216 


SMALL    HEAD.       TOP   AND    BOTTOM    BELTS 


X  > S  J  .Theoretical  line   \ 

'A       Column  ireed _/Y      of  in  flee 


Assumed  line      N 
of  inflection          *- 


1 

: 
4 

r| 

1C  in  pounds  p 

i  \  i  \  n  i  1  1  n 

erf  quart  foot 

M  1  i  i  1 

•s-l 

II 

null! 

j         ^^                        ~            Diagonal  steel  in  two  fayers         ...                 "*^>^^ 

^^  7 

i                      r              x      >'         / 

rectangular  sttel  in 
two  laytn 

\\ 

S 

g  la  pounas  per  linear  foot 


Fig.  57 


There  is  still  another  and,  if  possible,  more  serious  objection 
to  this  arrangement  of  rods  to  form  a  mat  or  double  layer  of  rods 
at  the  top  and  at  the  bottom  of  the  slab  near  the  columns..  This 
is  because  they  are  too  far  removed  from  each  other  in  the  slab 
for  the  elongations  of  the  steel  in  one  mat  to  be  resisted  by  lateral 
contractions  in  the  other.  The  reinforcement  does  not  therefore 
conspire  to  produce  the  slab  action  expressed  by  Poisson's  ratio, 
which  requires  that  the  interacting  steel  concerned  should  lie  approxi- 
mately in  the  same  zone  or  level. 

This  arrangement  is  illustrated  in  Fig.  57,  copied  from  Taylor 
and  Thompson's  Concrete  Plain  and  Reinforced,  p.  484.  In  this 
design  the  size  of  the  head  is  small  enough  to  reduce  the  width  of 
the  belts  so  greatly  that  not  only  are  the  areas  where  we  have  a 
single  layer  of  rods  on  the  plan  much  enlarged,  but  we  find  that 
nowhere  do  more  than  two  layers  lie  in  metallic  contact  with  each 
other,  and  the  areas  where  even  this  occurs  are  limited  to  one 
relatively  small  square  over  each  column,  and  one  of  equal  size 
at  the  middle  of  each  panel.  The  remaining  areas  are  subject  to 
the  law  of  single  rod  reinforcement,  where  we  must  assume  lateral 
action  to  be  such  as  greatly  to  diminish  K  for  the  combination,  a 
fact  very  injurious  to  the  efficiency  of  the  reinforcement.  This 
as  has  been  said,  is  due  partly  to  the  smallness  of  the  head  and 
partly  to  the  separation  of  the  layers  between  the  top  and  the 
bottom  of  the  slab. 


BAD    EFFECT   OF    ANY    SHARP    BEND    OR   ELBOW    IN    A   ROD 


217 


3rd.  Another  modification  of  design  without  change  of  ground 
plan  is  that  where  the  rods  that  are  carried  over  the  head  at  the 
top  of  the  slab  are  given  a  sudden  steep  dip  at  the  line  of  contra- 
flexure  to  carry  them  to  the  bottom  of  the  slab  at  that  line.  This 
is  also  illustrated  in  Fig.  57.  Such  sudden  bends  or  kinks  any- 
where in  the  rods  may  give  rise  to  very  serious  fractures  because 
of  straightening  out  under  tension,  especially  when  the  forms  are 
removed.  Such  bends  give  rise  to  great  differences  of  stress  in  the 
extreme  fibers  of  the  rods,  thus  diminishing  their  resistance  also. 
All  sudden  bends  in  rods  embedded  in  concrete  should  be  sedulously 
avoided  as  tending  very  effectively  to  crack  the  concrete,  whether 
the  rods  are  part  of  the  belts  or  in  the  frame-work  of  the  head,  as 
shown  in  Fig.  55,  in  which  are  many  such  angles  and  elbows  unsup- 
ported except  by  concrete,  and  therefore  objectionable. 

It  seems  fair  to  conclude  that  the  cracks  shown  in  the  plan 
of  the  floor  of  the  Deere  &  Webber  Company  Building,  Minnea- 
polis, tested  by  Mr.  Arthur  R.  Lord,  and  occuring  along  the  edges 
of  some  of  the  loaded  panels  at  the  upper  surface,  where  none  usually 
appear,  were  due  to  the  elbows  in  the  frame  work  of  the  head,  like 
that  in  Fig.  56,  in  conjunction  with  the  comparatively  small  resis- 
tance to  bending  in  a  vertical  plane  offered  by  the  rods  forming  this 
projecting  elbow. 

In  the  mushroom  head  the  only  bend  permitted  is  that  at  the 
elbow  of  the  radial  rods  where  a  strong  steel  neck  band  prevents 
any  such  bad  effect  as  has  just  been  pointed  out. 


Fig.  58 


218  TWO    WAY   REINFORCEMENT 

4th.  We  may  notice  a  form  of  design  in  which  the  diagonal 
belts  are  omitted  and  the  entire  panel  is  covered  by  rods  parallel 
to  the  sides  of  the  panel.  This,  while  apparently  very  different  in 
ground  plan  from  those  just  considered  does  not  differ  from  it 
materially  in  principle.  It  is  clear  that  the  lattice  pattern  of  the 
web  in  this  case  is  in  many  parts  of  the  panel  not  woven  so  close 
as  where  diagonals  exist, while  in  other  parts  of  the  mesh  the  num- 
ber of  layers  in  contact  with  each  other  has  been  decreased.  Experi- 
mental results  do  not  as  yet  enable  us  to  determine  with  certainty 
whether  Poisson's  ratio  for  this  combination  is  as  great  as  for  the 
mushroom.  Upon  that  depends  in  part  the  relative  efficiency 
of  the  two  arrangements.  A  form  of  this  design  is  seen  in  Fig.  58. 

The  maximum  deflections  at  the  center  of  a  loaded  panel  of 
the  system  of  Fig.  58,  would  occur  when  the  panels  touching  its 
four  sides  were  also  loaded.  In  this  particular  it  differs  from  a 
loaded  panel  in  a  mushroom  slab  which  would  theoretically  have 
its  deflection  slightly  decreased  by  loading  surrounding  panels, 
tho  this  is  too  insignificant  to  have  been  observed  as  yet. 

Deflections  shown  by  tests  of  this  system  of  two  way  reinforce- 
ment are  wholly  inconsistent  with  simple  beam  theory,  and  can 
only  be  explained  on  the  basis  of  slab  theory.  Nevertheless,  some 
of  its  advocates  attempt  to  design  its  reinforcement  and  com- 
pute its  strength  on  the  basis  of  beam  theory,  which  actual  de- 
flections show  to  be  untenable.  Such  attempts  should  be  entirely 
abandoned  as  erroneous  and  misleading. 

All  considerations  which  have  been  discussed  under  the  three 
previous  counts  are  to  be  taken  as  applying  equally  to  this  plan 
of  arranging  the  reinforcing  rods,  especially  as  to  carrying  of 
part  of  the  belts  thru  on  the  bottom  surface  at  columns. 

5th.  Another  element  of  design  is  the  relative  number  of 
rods  in  the  side  and  diagonal  belts.  We  have  previously  adduced 
reasons  to  show  that  in  a  square  panel  the  same  number  of  rods  is 
required  ultimately  in  the  diagonal  belts  as  in  the  side  belts,  tho 
for  stresses  less  than  the  yield  point  of  the  steel,  it  would  be  pos- 
sible to  diminish  the  number  of  rods  in  the  diagonal  belts  some- 
what. Equation  (34)  shows  that  for  equal  stresses  in  the  steel 
of  the  side  belts  the  number  of  rods  should  have  the  same  ratio 
as  the  lengths  of  the  sides. 

A  different  rule  from  this  has  been  erroneously  proposed, 
viz.,  that  the  ratio  of  the  number  of  rods  in  the  side  belts  should 
be  equal  to  the  ratio  of  the  cubes  of  their  lengths.  The  only  foun- 
dation for  this  rule  is  that  according  to  the  beam  strip  theory  as 


RELATIVE    CROSS   SECTION    OF   BELTS  219 

developed  in  Marsh's  Reinforced  Concrete,  p.  283,  a  rectangular 
plate  carried  by  a  level  rigid  support  around  its  perimeter,  would 
divide  the  load  per  unit  of  area  which  is  carried  by  two  unit-wide 
rectangular  strips  that  cross  each  other,  as  the  fourth  power  of  their 
lengths,  and  hence  would  carry  to  the  edges  of  the  rectangle  loads 
proportional  to  the  cubes  of  the  lengths  of  those  edges.  Were  this 
so,  the  case  of  a  horizontal  rigid  support  around  the  entire  peri- 
meter of  the  panel  is  wholly  different  from  support  on  columns 
at  the  corners,  and  such  a  rule  would  be  wholly  inapplicable  there- 
fore to  a  floor  slab  so  supported.  This  rule  was,  however,  evidently 
adopted  in  the  design  of  the  Larkin  Building,  Chicago,  as  shown 
by  a  -photograph  of  its  reinforcement  in  place  before  the  concrete 
was  poured,  to  which  the  writer  has  access  and  published  in  Cement 
Era  for  February,  1913.  The  very  exhaustive  tests  of  this  build- 
ing made  by  the  Concrete  Steel  Products  Company  of  Chicago, 
and  published  in  the  Cement  Era,  for  January  1913,  show  that  this 
ratio  of  rods  caused  the  stresses  for  the  larger  loads  to  be  more 
than  twice  as  great  at  the  middle  of  the  short  side  belts  as  at  the 
middle  of  the  long  side  belts.  This  was  assuredly  an  uneconomical 
distribution  of  steel,  since  correct  design  would  require  these  stresses 
to  be  equal,  when  in  fact  one  exceeded  the  other  by  120  to  140  per 
cent.  This  discrepancy  would  be  largely  rectified  by  making  the 
number  of  rods  directly  proportional  to  the  lengths  of  the  sides, 
as  required  by  (34). 

It  also  appears  that  the  diameter  of  the  mushroom  head  and 
the  width  of  belts  of  slab  rods  in  the  Larkin  Building  is  less  than  the 
limiting  size  in  the  standard  mushroom  system,  viz.  g  =  7(a+6)/16. 
This  makes  the  intersection  of  the  diagonal  belts  fall  nearer  the 
center  of  the  panel  than  the  edges  of  the  side  belts.  The  very 
considerable  effect  of  a  very  inconsiderable  change  of  this  width 
has  been  mentioned  on  p.  182.  The  result  would  be  that  the  steel 
would  for  this  reason  be  far  less  effective,  and  its  resistance 
would  be  more  nearly  in  accordance  with  (37)  than  with  (34),  a  loss 
of  perhaps  25  to  30%  in  its  effectiveness. 


220 


STIFFNESS  OF  MUSHROOM  SLAB  BRIDGES 


Tischers  Creek  Bridge,  Duluth 


Test  of  Tischers  Creek  Bridge  with  30  ton  construction  cars,  each  loaded  with  20  tons  of  rails 
Deflection  less  than  one  twenty  thousandth  part  of  the  span 


221 


CHAPTER  VI 

CALCULATIONS  OF  STRESSES  AND  DEFLECTIONS  VERIFIED 

BY  TESTS 

1.  Specimen  Computations  of  Stresses  and  Deflections  in 
Several  Slabs. 

Ford  Building,  Los  Angeles,  Calif. 
L!  =  28  x  12  =  336" 
L2  =  25'  10"  ==  310" 

Thickness  of  panel  tested  varied  from  llf  to  12f  inches.  Effec- 
tive thickness  taken  conservatively  at  1  If  inches  =  Li/28  =  L2/26.4 

By  (56)  C2  =  L2/Lj  =  0.92  nearly. 

Diameter  of  head   g=  7  (Ll  +  L2)/32  ==  141". 

Diameter  of  cap  Ll  —  B  =  0.2Ll  =  67".     R  =  O.SLi  =  268.8". 

Each  belt  of  twenty-five,  7/16"  round  rods. 

Cross  section  of  each  belt,  A  =  25    X  0.15+  =  3.76  sq.  inches. 

Depth  of  center  of  mid  side  belt  with  J  inch  concrete  covering 
dl  =  11.75  —  0.5  —  0.2  =  11". 

Depth  of  center  of  second  layer  of  slab  rods  at  panel  center, 
d2  =  11.75  —  0.5  —  0.64  =  10.6". 

Depth  of  bottom  surface  below  third  layer  of  slab  rods  at  edge 
of  cap  with  |"  covering,  d3  =  11.75  —  0.75  —  1.1  =  9.9". 

Design  load  per  square  foot  =  150  Ibs. 

Dead  load  per  square  foot  =  130  Ibs. 

Total  design  load,  W  =  280  X  28  X  25  5/6  =  202,550  Ibs. 

Test  load  W   =  300  X  28  X  25  5/6  =  217,000  Ibs. 

A  maximum  tension  is  found  in  the  slab  rods  at  the  middle  of  the 
long  side  belt,  and  is  to  be  computed  from  (34)  as  follows: 

217000  X  336 

/„  =    =  10,050  Ibs.  per  sq.  in. 

175  X   11    X  3.76 

Any  other  loading  within  elastic  limits  of  the  steel  would  produce 
proportionate  stresses. 

The  tension  in  the  steel  at  the  center  of  the  panel  is  computed 
by  (52),  as  follows: 

1.02  X  202550  X  336 

fa  =  =8184  Ibs,  per  so.  in. 

256  X  3.76  X  0.89  X  8.86 


THE    FORD    BUILDING,    LOS    ANGELES 

The  radial  tension  at  the  edge  of  the  cap  by  (44)  is  less  than 

217000  X  336  X  646  (3  X  1.61—1) 

=  19ooO  Ibs. 
800  X  9.9  X  3.76  X  310 

per  sq.  in. 

Hence  by  (45)  the  unit  compression  in  the  concrete  at  the  edge 
of  the  cap  is  less  than 

/c  =  19550  X  0.7/15  =  912  Ibs. 

The  compression  in  the  concrete  lengthwise  of  the  longer  side  belt 
at  its  middle  is  to  be  computed  from/s  and  (38)  as  follows,  by  tak- 
ing the  percentage  of  belt  reinforcement  at  0.24%,  the  corresponding 
value  of  i  =  0.76,  and  EB/EC  =  15: 

0.24  X    10050 

/0  =  —     =  211  Ibs.  per  sq.  in. 

0.76  X  15 

The  compression  at  the  center  of  the  panel,  where  the  percentage 
of  slab  reinforcement  may  be  conservatively  assumed  at  0.6%  and 
i  =  0.67,  may  be  computed  thus: 

/c  =  =   273  Ibs.  per  sq.  in. 

If  a  test  load  of  twice  the  design  load,  viz.,  in  this  case  of  300 
Ibs.  per  square  foot,  be  placed  upon  the  slab,  the  deflections  which 
will  be  produced  by  the  addition  of  this  total  load  of  217,000  Ibs. 
may  be  computed  as  follows: 

217000  x  3363 

By  (54),      A  z,  =  -  =  0.169" 

10.7  x!010x  II2  x  3.76 

0.92  x  217000  x  3363 

By  (55),      A  z2  =  -  =  0.268" 

6.56  xl010xl0.62x  3.76 

217000  x  3363  x  2.084 

By  (58),      Az3=-  =  0.0485" 

96  x  1010  x  9.92  x  3.76 

0.92  x  217000  x  3363 

By  (60),      A  z4  =  -  =  0.101" 

20xl010x9.92x3.76 

By  (61),  Dl  =  0.22  inch,  and  D2  =  0.37  inch. 
The  observed  deflection  was  f  in.    =   0.375  in.     The  deflection 
computed  by  the  less  exact  equation  (71)  is  D2  —  0.378. 

Dl  1  D2  1 

=        — ,     and  = 

LI  1530  VL  2    ,    L  2 


TISCHER'S  CREEK  BRIDGE  223 

Any  loading  differing  from  this  would  produce  deflections 
proportionate  to  its  intensity. 

The  observed  results  of  quite  a  number  of  tests  of  mushroom 
slab  floors  are  to  be  found  on  pp.  32  and  44  of  Turner's  Concrete 
Steel  Construction.  These  are  there  compared  with  results  com- 
puted according  to  Turner's  empirical  formula,  which  translated 
into  our  present  notation  has  been  reproduced  in  equation  (72). 
The  observed  and  computed  results  show  a  very  close  agreement. 
The  results  given  by  (72)  are  in  close  agreement,  as  has  been  seen, 
with  those  derived  from  (61). 

Some  of  these  test  slabs  present  peculiarities  of  reinforce- 
ment such  as  need  to  be  individually  considered  in  order  to  make 
exact  computations  of  their  deflections.  It  is  thought  that  the 
specimen  computation  already  given  will  afford  sufficiently  guidance 
in  the  methods  to  be  employed. 

Having  considered  the  stresses  and  deflections  of  a  slab  which 
is  near  the  minimum  thickness  for  the  standard  mushroom  system, 
viz.  Z/i/35,  it  will  be  instructive  to  consider  a  specimen  or  two 
near  the  maximum  thickness  Lj/20. 

The  Bridge  Over  Tischer's  Creek,  Duluth,  Minn.,  is  such  an 
example.  See  cuts  opposite  page  157  and  page  220.  It  is  supported 
on  three  rows  of  columns  crossing  the  gorge,  at  a  distance  apart  of  27 
feet  center  to  center  of  columns,  the  two  street  car  tracks  being  over 
the  side  belt  that  lies  along  the  center  line  of  the  bridge  lengthwise. 
Each  of  these  rows  consist  of  six  columns  lengthwise  of  the  bridge, 
at  a  distance  apart  of  26  feet  from  center  to  center,  so  that 

L!  =  27  x  12  =  324" 
L2  =  26  x  12  =  312" 

The  size  of  the  mushroom  heads  and  width  of  the  belts  is  12  feet, 
which  is  in  excess  of  7  (Lx  +  L2)/32  ==  139  1/8"  ==  11.6',  thus  giv- 
ing great  stiffness.  The  object  to  be  obtained  by  maximum  thick- 
ness and  large  head  is  to  secure  great  stiffness  and  so  reduce  vib- 
rations as  well  as  decrease  deflections.  There  are  twenty  9/16 
inch  round  slab  rods  in  each  belt,  or  a  total  cross  section  in  each 
belt  of  AI  =  5  square  inches  of  metal.  The  slab  is  15"  deep  at 
its  thinnest  part  at  the  gutter  on  each  side  of  the  roadway,  and 
the  steel  is  kept  down  to  that  level  throughout  the  slab,  altho  at 
the  crown  of  the  roadway  under  the  tracks  and  over  the  center 
row  of  columns  the  slab  is  5"  thicker,  or  20",  with  the  same  thick- 


224 


TISCHER  S    CREEK    BRIDGE 


ness  over  the  side  rows  of  columns  where  the  sidewalks  are.  The 
mean  thickness  is  somewhat  in  excess  of  L2/20.  This  makes 
di  =  19"  for  the  short  side  belts,  dl  =  17"  for  the  long  side  belts 
and  rf:i  =  14"  approximately  for  the  heads.  The  design  load  per 
square  foot  =  150  pounds.  The  dead  load  of  the  slab  per  square 
foot  ==  300  pounds.  Hence  W  =  450  x  26  x  27  =  315,900  pounds. 
The  effective  cross  section  of  slab  steel  is  so  great  by  reason  of  large 
heads  that  instead  of  (34)  we  may  take 


W  L 

200  d^  Al 


(34)' 


For  the  long  side  belt  this  gives  fs  =  6,033  pounds  per  square  inch. 
The  total  load  imposed  on  the  slab  might  be  made  six  times  as  great 
without  causing  the  steel  to  reach  its  yield  point,  and  the  live 
load  might  become  900  pounds  per  square  foot  without  causing  /s 
to  exceed  16,000  pounds. 

This  slab  was  tested  as  shown  in  the  cut,  page  220,  by  running 
two  construction  cars  loaded  with  20  tons  of  rails  each  over  the 
bridge  at  the  same  time  along  one  track  of  the  short  side  belt  26 


I  I 


VIEW  OF  REINFORCING  STEEL 
Flat  Slab  Bridge,  Denver,  Colo.          Spans  43  ft.  6  in.  Carries  Heavy  Interurban  Cars 


CURTIS    STREET    BRIDGE,    DENVER  225 

feet  long.  Weight  of  each  car  =  60,000  pounds.  Weight  of  rails 
40,000  pounds.  Total  weight  of  train  =  200,000  pounds  extend- 
ing over  several  spans.  The  deflections  were  too  small  to  be  dis- 
covered by  observations  with  level  and  rod.  It  is  useless  to  attempt 
to  compute  the  deflection  of  this  slab  under  the  test  load  because 
the  four  steel  rails  of  the  railway  tracks  across  the  bridge  were  so 
fastened  to  the  steel  cross  ties  which  were  embedded  in  the  con- 
crete as  to  make  the  rails  a  part  of  the  reinforcement  of  the  slab. 
They  furnish  a  cross  section  of  reinforcement  equal  perhaps  to 
7  AI,  which  would  effectually  bar  the  application  of  our  deflection 
formulas  and  reduce  deflections  to  very  small  quantities. 

In  so  thick  a  slab  as  this  the  action  of  any  contemplated  load 
is  widely  distributed  by  the  slab  itself,  and  such  loads,  as  well  as 
all  shocks  and  vibrations  are  largely  dissipated  or  absorbed  by  the 
body  of  slab  itself  without  causing  observable  local  stresses  as  they 
do  in  steel  structures. 

The  Curtis  Street  Bridge,  Denver,  Colorado,  is  one  of  four 
bridges  across  Cherry  Creek,  shown  by  the  cut  on  page  224,  con- 
structed on  the  mushroom  system.  It  has  three  rows  of  three 
columns  each  crossing  the  stream,  the  middle  column  of  each  row 
in  mid  stream  with  spans  of  42  feet  between  columns  centers  length- 
wise of  the  bridge,  thus  obstructing  the  waterway  as  little  as  pos- 
sible. It  has  a  width  of  28  feet  between  column  centers.  The 
slab  is  17  inches  thick  at  the  gutters,  26.5  inches  at  the  sidewalks 
outside  the  gutters,  and  21"  over  the  center  row  of  columns.  The 
sidewalk  is  stiffened  with  fourteen  3/8"  round  rods  lengthwise 
just  below  its  top  surface  as  supplementary  reinforcement,  and 
there  is  an  outside  parapet  giving  added  stiffness.  There  are 
also  three  stiffening  rods  24"  apart  across  the  bridge  midway 
between  columns.  There  are  three  ring  rods,  and  the  width  of  the 
belts  is  16'.  This  is  in  excess  of  7  (Ll  +  L2)/32  =  183.75"  - 15  5/16'. 
The  heads  are  exceptionally  stiff  each  having  twelve  1  3-8"  round 
radial  rods.  Each  belt  has  twenty-six  5/8"  round  rods,  hence 
AI  =•  26  x  0.3  =  8  square  inches  nearly. 

L!  ==  42  x  12  =  504"    ,     L2  =  28  x  12  =  336". 

The  dead  load  per  square  foot  =  300  pounds. 

The  design  load  per  square  foot  =   150  pounds. 

W        450  x  42  x  28   ==   529,200  pounds. 

rfi   ==   20"  for  long  side  belt. 

Compute  the  stress  in  the  steel  by  (34)  modified  to  (34) '  by 
reason  of  exceptional  stiffness,  and  we  obtain /s  =  13,320  pounds. 


226  HIGH    DEGREE    OF    SECURITY    IN    SLABS 

Compute  the  central  deflection  due  to  a  test  load  of  100  pounds 
per  square  foot.  Let  d3  =  16".  Then  in  (71)  L2/L1  =  2/3:  hence 
C2=  3/4,  and  we  have  D2  =  0.125".  This  is  probably  considerably 
in  excess  of  the  correct  deflection,  since  the  slab  is  stiffer  than  the 
one  considered  in  equation  (71),  which  was  derived  for  20  foot  spans. 
More  correct  values  are  to  be  computed  from  (54),  (58)  and  (61). 
Moreover  for  such  comparatively  light  stresses  in  the  concrete, 
the  deflections,  as  we  have  seen  previously  fall  short  of  those  com- 
puted by  the  formula,  which  agrees  with  experiment  for  stresses 
nearer  the  yield  point  of  the  steel.  D2  =  0.125"  is  less  than 
one  four-thousandth  of  the  span,  and  the  deflection  under  the 
working  load  would  undoubtedly  be  less  than  one  sixth-thousandth 
of  the  span. 

A  word  is  here  in  place  respecting  working  stresses  and  the 
factor  of  safety  in  the  reinforcement  of  slabs,  to  the  effect  that 
the  same  values  of  these  quantities  in  slabs  affords  a  greater  degree 
of  security  than  in  ordinary  structural  steel  construction,  and  that 
occurs  for  several  reasons: 

1st.  Steel  rods  such  as  are  used  in  slabs  have  a  higher  yield  point 
by  perhaps  25%  than  the  steel  of  other  structural  members.  Fur- 
thermore, it  is  quite  possible  and  desirable  to  use  a  higher  carbon 
steel  for  these  rods  than  the  mild  steel  necessarily  used  in  structural 
work,  where  it  must  be  manipulated  in  such  ways  that  high  carbon 
steel  cannot  be  used.  But  in  these  rods  which  suffer  no  usage 
tending  to  impair  their  condition,  there  is  good  reason  to  use  a  steel 
of  higher  yield  point  and  greater  ultimate  strength.  This  yield 
point  may  readily  be  70%  greater  than  that  of  ordinary  mild  steel 
for  structural  purposes. 

2nd.  Rods  embedded  in  concrete  do  not  yield  as  do  bare 
single  rods  in  a  testing  machine  or  elsewhere  by  the  formation  of 
a  neck  and  drawing  out  at  that  point.  The  concrete  embedment 
prevents  that. 

3rd.  In  a  reinforcement  consisting  of  multiple  parallel  rods 
acting  together,  no  single  rod  can  become  overstrained  and  yield  to 
any  appreciable  extent  before  bringing  into  play  adjacent  rods. 
This  makes  the  construction  tough,  and  not  liable  to  sudden  col- 
lapse, as  well  as  obviates  concentration  of  stresses  thus  ensuring 
a  high  degree  of  security. 

2.  Further  Calculations  of  Test  Slabs.  A  verification  of  the 
calculated  stresses  and  deflections  by  discussion  of  test  data  in  case 
of  several  slabs  appeared  in  the  Trans.  Am.  Soc.  C.  E.,  for  1914, 


NORTHWESTERN    GLASS    COMPANY    BUILDING  227 

Vol.  LXXVII,  page  1338  to  page  1453.  Some  of  the  more  important 
calculations  and  verifications  are  here  inserted,  but  for  details  consult 
the  original  paper. 

Test  of  the  Northwestern  Glass  Company  Building,  Minneapolis, 
made  May,  1913,  by  F.  R.  McMillan. 

L!  -  17  X  12  ==  204".  L2  ==  16  X  12  =  192".  Rough  slab 
8"  thick. 

Diameter  of  cap  =  50";  diameter  of  head  =  87". 

Each  belt  consisted  of  fifteen  |"  round  rods  with  no  other  laps 
than  merely  splices.  Standard  Mushroom  heads. 

Design  live  load  400  pounds  per  square  foot. 

Test  load  taken  to  be  equivalent  to  a  total  panel  load  of  200,000 
pounds  uniformly  distributed. 

The  unit  stress  at  mid  span  of  the  shorter  side  belts  by  equation 
(34),  page  183,  is 

200,000  X  16  X  12 

L  =  •    =  18000  Ibs.  per  sq.  inch. 

185  X  7.31  X  1.6567 

Two  of  the  observed  readings  were  17000  pounds,  and  it  is  prob- 
able that  if  all  the  panels  instead  of  only  four  of  the  slab  had  been 
loaded  at  once  each  with  a  uniformly  distributed  load  of  200,000 
pounds,  all  the  rods  in  these  side  belts  would  have  shown  nearly  or 
or  quite  the  foregoing  computed  values  of /s. 

The  observed  stresses  at  mid  span  of  the  longer  side  belts  in  this 
test  were  somewhat  less  than  at  mid  span  of  the  shorter  belts,  a 
divergence  from  (34)  which  seems  to  be  due  to  the  effect  of  the  wall 
support  on  the  long  side  of  the  wall  panels. 

By  equation  (52),  page  196,  the  unit  stress  in  the  middle  rod  of 
the  diagonal  belt  at  the  center  of  the  panel  is 

200,000  X   204 

f,  =  =  l£>,o70  Ibs.  per  sq.  inch. 

256  X  0.89  X  6.94  X  1.6567 

The  observed  unit  stress  at  center  of  interior  panel  =  14,200  Ibs. 
'which  was  larger  than  in  the  rods  on  either  side  of  it,  as  required  by 
theory,  but  the  unit  stress  at  the  center  of  one  of  the  wall  panels  was 
20,500  Ibs.,  probably  due  to  the  fact  that  there  is  less  cantilever 
support  at  the  wall  than  is  exerted  by  interior  column  heads. 

The  greatest  stresses  occur  at  the  edge  of  a  cap  when  the  four 
panels  around  it  are  all  loaded.  Such  a  distribution  of  load  occurred 
in  this  test  when  there  was  a  load  on  each  of  the  four  panels  which 


228  DEERE    AND    WEBBER    COMPANY    BUILDING 

was  assumed  to  be  equivalent  to  106,250  Ibs.  The  limiting  unit- 
stress  at  the  edge  of  the  cap  calculated  by  Equation  (44)  is  20,700 
Ibs.,  on  the  side  belt  and  the  similar  equation  for  the  diagonal  belt 
gives  19,500  Ibs. 

The  two  highest  unit  stresses  observed  at  the  edge  of  the  central 
cap  were  in  the  diagonal  belt  on  opposite  edges  of  the  cap  and  were 
17,500  Ibs.  and  20,000  Ibs.  respectively,  There  was  also  an  abnormal 
unit  stress  of  22,400  Ibs.  in  a  middle  diagonal  rod  of  a  cap  at  the  edge 
of  the  loaded  area  due  to  a  forcible  bending  of  a  radial  rod. 
By  equation  (71)  the  deflection  in  the  diagonal  center  of  the  panel  is 

200,000  X  16  X  (17)2  X  1728 

JL/O  —  —  — —  =  0.4:ZZ  in. 

4.75  X  1010  X  1.6567  (6.9S75)2 

While  this  load  rested  on  the  slab  for  18  hours  the  deflection  grad- 
ually increased  from  0.416  in.  to  0.456  in.,  so  that  the  calculated 
deflection  is  entirely  satisfactory. 

By  Equations  (61),  (54)  and  (58),  as  corrected  by  the  intro- 
duction of  96  in  place  of  60  the  values  of  the  deflections  at  mid 
span  of  the  side  belts  are : 

At  point  12:  Computed  0.229  in.:  Observed  0.22  in. 

At  point  20:  Computed  0.228  in.:  Observed  0.20  in. 

The  observed  side  deflections  of  a  single  loaded  panel  are  neces- 
sarily less  than  where  surrounding  panels  are  equally  loaded. 

The  deformations  and  stresses  observed  in  the  vertical  steel  in 
one  of  the  columns  at  a  corner  of  a  panel  loaded  with  some  800  Ibs. 
per  sq.  foot,  have  by  careful  analysis  given  a  probable  limiting  value 
of  that  part  of  the  unbalanced  moment  due  to  this  load  which  was 
resisted  by  the  column  itself.  The  diameter  of  the  column  core  was 
27  inches  and  the  column  was  continued  to  the  upper  stories.  An- 
alysis shows  that  of  the  total  unbalanced  moment  W  L/12  which 
is  to  be  resisted  at  a  support  only  about  one  fifth  took  effect  upon 
the  column  so  far  as  shown  by  differences  of  the  deformations  of 
steel  and  concrete  in  the  opposite  sides  of  the  column. 

Test  of  the  Deere  and  Webber  Company  Building,  Minneapolis,* 
made  by  A.  R.  Lord,  Nov.  1910. 

Design  live  load  225  Ibs.  per  square  foot. 

L!  =  19'1"  =  229  inches.     L2  =  18'8"  =  224  inches. 

Slab  9  and  3/8  inches  thick;  concrete  40  days  old  at  beginning 
of  test.  All  slab  rods  7/16  inch  round,  wilh  12  rods  in  each  side 


*Reported  by  Mr.  Lord.     Proceedings  Nat.  Assoc.  Cement  Users,   Vol.  VII 
Philadelphia,  1911. 


DEERE    AND    WEBBER    COMPANY    BUILDING  229 

belt  and  15  in  each  diagonal;  all  slab  rods  7J  inches  between  centers, 
so  that  the  side  belts  may  be  taken  to  be  80  inches  wide  and  the 
diagonal  nearly  90  inches.  The  head  was  a  rectangular  diamond 
frame  with  four  rods  extending  entirely  across  it.  Column  caps 
54  inches  in  diameter.  Mean  size  of  cap  =  0.24  L. 

Take  dl  =  8.5  inches,  d2  =  8  inches  and  c?3  =  7.6  inches.  The 
calculated  unit  stress  at  mid  span  on  the  short  side  by  equation 
(34)  under  the  test  load  of  350  Ibs.  per  square  foot  or  a  total  load  of 
350  X  356  2/9  ==  124,678  Ibs.  is 

_  _   _124678_X22i_      _  =  ^ 

175  X  8.5  X  12  X  0.15 

The  observed  unit  stress  at  one  mid  span  was  10,400  Ibs.  and 
at  others  somewhat  less  than  10,000,  so  that  this  is  a  satisfactory 
determination  of  the  greatest  stress  at  mid  span  of  the  short  side. 

Equation  (34)  gives  for  the  unit  stress  at  mid  span  of  the  longer 
side  10,220  Ibs.  The  mean  observed  result  was  only  6600  Ibs.  There 
was  some  reason  in  the  arrangement  of  loaded  and  unloaded  panels 
to  expect  a  result  of  this  kind,  tho  larger  stresses  would  hardly  be 
expected  on  the  short  side  than  on  the  long  side  unless  the  bulk- 
head cracks  in  the  slab  had  a  considerable  influence. 

The   unit   stress   at   the   panel   center   calculated    by    equation 


124678  =74401,,, 


256  X  0.89  X  14  X  0.15  X  8 

This  is  larger  than  the  mean  of  four  observed  values,  and  larger 
than  all  but  one  of  them  which  has  an  appreciably  abnormal  observed 
value. 

The  limiting  unit  stress  at  the  edge  of  the  cap  by  equation  (44) 
in  case  we  assume  a  mean  of  13  rods  to  a  belt  is 

124678  X  229  X  453          /3  X  52441 


/ 
V 


800  X  7.6  X  13  X  0.15  X  224  V      30625 

The  largest  observed  unit  stress  in  a  side  belt  was  20,000  Ibs.  A 
similar  computation  for  the  diagonal  belt  gives  22440  Ibs.  There 
were  two  observed  stresses  larger  than  this,  one  of  23400  Ibs.  and  one 
of  24,200,  due  no  doubt  to  the  lighter  head  used  in  this  construction. 
Applying  Equation  (71)  to  the  calculation  of  the  deflection  at 
the  panel  center  we  have 

124678  X  224  X  229  X  229 

Z>9  =  -  -   =  0.23  inches. 

4.75  X  1010  X  64  X  14  X  0.15 

The  mean  value  of  seven  readings  was  0.224. 


230  THE    LARKIN    BUILDING    CHICAGO 

The  mean  deflections  of  adjacent  panels  were  0.291,  0.271,  or 
0.306  inches,  these  all  being  somewhat  larger  because  adjacent  panels 
were  not  loaded. 

The  interaction  of  contiguous  panels  across  the  column  heads 
was  evidently  of  considerable  amount  in  this  slab,  which  differed 
from  mushroom  construction  in  having  no  stiff  connection  with 
columns  such  as  is  afforded  by  elbow  rods.  Where  the  greatest 
stresses  in  steel  and  concrete  occurred  around  the  cap  there  appears 
to  be  a  deformation  ratio  ec/ea  of  0.5  to  0.6. 

The  Larkin  Building,  Chicago,  tested  by  A.  R.  Lord.*  The 
design  of  the  belt  reinforcement  is  of  the  usual  four-way  type  em- 
ployed in  the  Mushroom  system,  but  the  column  heads  omit  much  of 
the  steel  and  in  place  of  it  use  a  depressed  head  or  "drop"  to  resist 
shear  and  flexure,  which  is  8  ft.  square  and  6.75  inches  thick,  and 
is  placed  on  top  of  each  column  cap  under  the  slab  and  integral 
with  it.  The  panels  are  24'2"  by  20',  or  Ll  =  290"  and  L2  =  240." 
The  thickness  of  the  slab  is  9  inches  except  at  the  drop  where  it  is 
15.75  inches.  The  diameter  of  the  cap  =  60  inches.  The  width 
of  side  belts  is  90  to  93  inches  and  diagonals  105  to  108  inches.  All 
belt  rods  are  \  inch  rounds,  13  in  each  short  side  belt,  22  in  each 
long  side  belt  and  21  in  each  diagonal  belt. 

di  =  8  inches,  d2  =  7.75  inches,  d3  =  14  inches. 

The  floor  was  designed  for  a  dead  load  of  about  120  Ibs.  per  square 
foot  and  a  live  load  of  from  225  to  250  Ibs.  with  a  maximum  test 
load  of  twice  the  sum  of  these,  or  actually  739  Ibs.  per  square  foot. 

The  total  panel  load  producing  stress  was 

W  =  738  X  20  X  24  1/6  =  356,700  Ibs. 

By  (34)  the  unit  stress  at  mid  span  of  the  shorter  side  belt  is 
/8  -  24,000  Ibs.  The  observed  stress  was  24,200  Ib.  between  two 
loaded  panels,  which  would  be  decreased  slightly  if  adjacent  panels 
were  equally  loaded  as  contemplated  in  equation  (34). 

For  the  long  side  belt  with  22  rods  (34)  gives  the  computed  unit 
stress  /8  =  17,000.  No  observed  value  was  so  large  as  this,  because 
all  long  sides  were  at  the  edge  of  the  loaded  area. 

By  (52)  the  calculated  value  of  the  unit  stress  in  the  middle 
rod  of  the  diagonal  at  the  panel  center  is/s  =  14,070  Ibs.  The  ob- 

*  The  results  of  this  test  were  presented  by  Mr.  Lord  to  the  Ninth  General 
Convention  of  Cement  Users.  Extracts  from  his  paper  appeared  in  the  Cement 
Era,  Jan.  1913,  page  53. 


THE    LARKIN    BUILDING    CHICAGO  231 

served  value  is  stated  as  12,900  which  is  probably  a  mean  stress  in 
the  rods  of  the  diagonal  belt  and  perhaps  less  than  the  stress  in  the 
middle  rod. 

To  find  the  unit  stress  in  the  steel  at  the  edge  of  the  cap,  it  is 
evident  that  owing  to  the  drop  the  effect  of  the  cap  in  causing  abnor- 
mal stresses  at  its  edge  will  be  much  reduced.  Assume  that  the 
stress  will  not  exceed  that  obtained  from  (43)  at  the  column  center 
where  B  =  Llj  then 

_  W  L!  (Li/L2  +  1)   _       356,700  X  290  X  2.21        =  1Q  g()()  ^ 
400  rf3  Al  400  X  14  X  19.25  X.  19635 

in  which  a  mean  belt  of  19.25  rods  is  assumed  for  purposes  of  com- 
putation because  all  the  belts  are  intimately  combined  in  their 
action  in  the  head.  Under  this  test  load  no  one  of  the  columns 
was  completely  surrounded  by  loaded  panels,  and  this  computed 
stress  would  be  approached  only  very  exceptionally.  There  was  one 
observation,  however,  of  10,400  Ibs.  and  others  of  7300  Ibs.  and 
7000  Ibs. 

If,  moreover,  we  suppose  that  no  noticeable  abnormal  action  is 
to  be  looked  for  at  the  edge  of  the  drop,  then  the  stress  would  be 
calculated  by  (43)  by  taking  B  =  290  —  96  ==  194",  and  d3  =  7", 
with  a  mean  belt  of  19.25  rods  which  gives  /s  =  14,400  Ibs.  The 
largest  observed  unit  stress  in  any  rod  at  the  edge  of  the  drop  was 
14,500  Ibs.  in  a  side  belt  and  14,200  Ibs.  in  a  diagonal  belt. 

The  test  load  for  deflections  was  618  Ibs.  per  square  foot.     Hence 
W    =  618  X  24  1/6  X  20  =  298700  Ibs. 

By  (54)  A  *  2f  7Q°  (29°)3  -  =  0.2463". 

6.56  X  1010  (7.75)2  X  21  X  .19635 

By  (55)  A  „   = 298TOOX240J290f =  ^^ 

6.56  X  1010  (7.75)2  X  21  X  .19635 

9Q&7On  f9Qn"l3  9  91 

Bv  (58)  A  z,  =  =  0.0188". 

96  X  1010  X  142  X  22  X  .19635 


298700  X  240  (290) 


By  (60)  A  z4  =  =  0.0375 

20  X  1010  X  142  X  21  X  .19675 

By  (61)  Dl  =  0.265"  and  D2  =  0.408". 


-// 


232 


THE  ST.  PAUL  BREAD  COMPANY  BUILDING 


The  corresponding  observed  values  were  0/26"  and  0.40",  res- 
pectively, an  agreement  which  is  surprising  in  view  of  existing  cir- 
cumstances as  to  reduction  of  steel  in  the  head  and  displacement  of 
lines  of  inflection  in  this  slab  compared  with  the  Mushroom  con- 
struction for  which  the  equations  were  deduced. 

The  St.  Paul  Bread  Company  Building.  The  floor  of  the  St. 
Paul  Bread  Company  Building  is  a  rough  slab,  6  in.  thick,  and  has 
16  by  15  ft.  panels,  reinforced  with  f  in.  round  rods,  ten  in  each  belt. 
The  design  load  was  100  Ib.  per  sq.  ft.  The  slab,  Fig.  59,  was  con- 
structed in  winter  and  fro/en,  but,  as  the  final  test  was  postponed 
until  August,  1912,  the  slab  was  very  fully  cured,  considerably  more 
so,  in  fact,  than  most  slabs  when  subjected  to  test.  The  test  was 
made  by  Professor  W.  H.  Kavanaugh,  of  the  University  of  Minnesota, 
in  the  following  manner: 


6  'Rough  Slab. 


10-%"Rocls. 


Face  of  Slab  to 
Rod  Center 

1.— 15/16          11.—  11/10 
•2.-1:Wo         12.-^",, 

5.-18Ac        15.-  lV« 


•is'o- 


DIAGRAM  SHOWING  LOCATION  OF  TEST   POINTS  IN  FLOOR  SLAB. 

SAINT  PAUL  BREAD  COMPANY  BUILDING. 

"TURNER  MUSHROOM  SYSTEM1' 

Fig.  59 


THE    ST.    PAUL    BREAD    COMPANY    BUILDING  233 

First,  extensometer  measurements  were  made  on  seventeen  8  in. 
lengths  of  slab  rods,  which  were  exposed  under  a  single  loaded  panel, 
three  of  these,  Nos.  1,  2,  and  3,  being  at  the  middle  of  three  rods  of 
one  long  side  belt  at  the  edge  of  the  load,  and  the  remaining  fourteen 
distributed  over  the  central  area  of  one  diagonal  belt.  Second, 
measurements  of  deflections  were  made  at  two  points,  one  at  the 
center  of  the  panel  and  one  near  the  middle  of  the  interior  edge  of 
one  long  side  belt.  Third,  the  embedment  of  the  rods  was  tested. 
Table  1  contains  the  observed  elongations  due  to  change  of  loading 
at  all  the  seventeen  positions  for  each  of  the  test  loads,  108.4,  316.8, 
and  416.8  Ibs.  per  sq.  ft.,  as  well  as  after  the  removal  of  the  load. 
A  comparison  of  the  observed  elongations  at  symmetrical  points 
reveals  such  discrepancies  in  the  observations  as  to  require  some 
preliminary  discussion. 

Observations  Nos.  5  and  6  were  on  a  pair  of  diagonal  rods  on  each 
side  of  and  adjacent  to  the  diagonal  line  of  the  panel,  there  being  no 
rod  exactly  on  the  diagonal,  and  situated  just  beyond  the  edge  of 
the  other  diagonal  belt.  No  reason  can  be  discerned  for  any  differ- 
ence between  these  elongations,  but  the  wide  difference  that  appears 
must  be  due  to  some  peculiarity  in  one  of  the  rods,  such  as  a  crook 
or  bend,  or  some  lack  of  homogeneity  in  the  concrete.  Comparing 
Observations  Nos.  5  and  6  with  Nos.  4,  10,  15,  and  16,  which,  being 
at  about  the  same  distance  from  the  center,  should,  by  Equation 
(49),  have  about  the  same  elongations,  it  is  found  that  No.  5  is 
abnormally  large,  and  at  the  same  time  No.  16  is  abnormally  small. 
No.  8  is  another  set  of  abnormal  results,  which  is  evident  from  the 
fact  that,  being  midway  between  Nos.  4  and  11,  its  elongations  should 
lie  between  them;  it  is  larger  than  it  should  be,  with  a  final  com- 
pression after  removing  the  load.  Nos.  7  and  17  should  be  the 
same,  and  Nos.  4  and  17  should  be  alike.  The  varying  embedments 
of  the  portions  of  this  rod  which  were  observed  show  that  there 
was  probably  a  kink  in  it,  which  might  account  for  the  observed 
discrepancies.  It  is  possible,  however,  that  some  such  differences 
may  appear  when  the  loading  is  piled  on  one  side  of  the  panel  before 
piling  it  on  the  other.  No  such  explanation,  however,  will  fit  the 
case  of  Nos.  12  and  13,  which  are  at  the  middle  of  the  two  rods  ad- 
jacent to  the  diagonal  line  at  the  center  of  the  panel.  There  appears 
to  be  no  question  that  No.  13  is  abnormally  large,  for  No.  12  agrees 
well  with  others,  being  only  a  little  larger  than  those  at  the  nearby 
positions  9.  11,  and  14,  and  the  values  at  No.  13  are  in  wide  dis- 
agreement with  them.  The  very  considerable  differences  between 
results  which  should  apparently  be  equal  makes  it  evident  how  in- 


234 


THE  ST.  PAUL  BREAD  COMPANY  BUILDING 


TABLE  1 

OBSERVED    ELONGATIONS,    IN    INCHES    PER    MILLION    INCHES 
UNDER  GIVEN  LOADS  PER  SQUARE  FOOT 

Note. — Obtain  actual  unit  stresses  by  multiplying  by  30. 


Observation 
No. 

Elongati< 
108.4 

1 

255 

2 

236 

3 

244 

4 

63 

5 

64 

6 

66 

7 

45 

8 

218 

9 

89 

10 

68 

11 

122 

12 

153 

13 

276 

14 

60 

15 

74 

16 

11 

17 

34 

Elongations  under  the  Following  Loads,  in  Pounds. 


316.8 


500 
473 
464 
207 
262 
168 
152 
370 
266 
120 
263 
327 
534 
204 
164 
40 
70 


416.8 

0 

598 

80 

549 

3 

572 

-  57 

283 

145 

387 

164 

271 

114 

220 

69 

421 

-  64 

372 

152 

159 

46 

347 

146 

400 

28 

653 

93 

282' 

80 

165 

18 

22 

-  30 

124 

17 

exact  single  determinations  must  often  be  by  reason  of  bends  in  the 
rods,  lack  of  homogeneity  in  the  concrete,  etc.,  and  emphasizes  the 
importance  of  carefully  laying  belt  rods  straight  and  having  them 
spaced  uniformly,  as  well  as  embedded  equally,  before  pouring  the 
concrete,  if  consistent  results  are  desired.  It  also  shows  the  impor- 
tance of  checking  all  readings  by  readings  at  symmetrical  positions. 

It  may  be  stated  in  general  that  the  observed  unit  stresses  and 
the  deflections  in  this  test  are  less  than  they  would  be  for  a  slab 
tested  at  the  stage  of  curing  at  which  tests  are  usually  made,  a  stage 
to  which  the  equations  apply  more  precisely.  In  consequence  of 
this,  all  the  computed  results  will  exceed  to  some  extent  those  actually 
observed. 

Apply  Equation  (34)  to  compute  the  unit  stress  at  Nos.  1,  2,  and 
3  of  the  long  side  belt.  Assuming  dl  =  5.3  in., 

100,000  X  192 

L   =    -  -  =  18,800  Ib.  per  sq.  in. 

175  X  5.3  X  1.1 

in  case  of  many  panels  equally  loaded.  The  mean  observed  unit 
stresses  for  three  rods  at  the  edge  of  the  load  was  17,190  Ibs.,  the 


THE    ST.    PAUL    BREAD    COMPANY    BUILDING  235 

stress  in  each  of  the  rods  being  practically  the  same,  a  fact  that 
speaks  well  for  the  stiffness  of  the  head.  The  computed  unit  stress 
at  the  center  of  the  panel  is 

100,000  X  192 

fs  =  =  1  o,l 50  Ib.  per  sq.  in. 

246  X  0.9  X   5  X   1.1 

The  observed  unit  stress  at  No.  12  was  12  000  11).,  tho  at  No.  13 
the  abnormal  value  of  19  600  Ib.  was  found. 

As  the  observations  of  stresses  and  deflections  were  made  when 
only  a  single  panel  was  loaded,  and  the  computations  assume  that  all 
the  panels  are  equally  loaded,  any  very  close  agreement  of  Table  1 
with  the  observed  results  is  not  to  be  expected;  nevertheless,  com- 
parison with  that  table  shows  that  the  computed  results  agree  with 
the  observations  far  better  than  the  observations  agree  among  them- 
selves at  such  symmetrical  points  as  admit  of  any  comparison. 

The  deflection  at  the  center  of  the  panel  under  a  load  of  more 
than  double  the  design  load  plus  once  the  dead  load,  namely,  316.8 
Ib.  per  sq.  ft.,  was  0.32  in.,  which  is  less  than  1/800  of  the  diagonal 
span.  To  compute  the  deflections  at  the  panel  center,  apply  Equa- 
tion (71),  as  follows: 

100  000  X  180  X  1922 
D.>  =  =  0.50  in. 

4.75  X  101()  X  52  X  1.1 

Computation  of  the  deflection  at  the  point  near  the  middle  of  the 
interior  edge  of  a  long  side  belt  gives  a  deflection  for  a  load  of  100  000 
Ib.  on  each  panel  of  approximately  0.4  in. 

TABLE  2 

Deflections,  in  Inches,  Under  Given  Loads  per  Square  Foot 

Test  Loads,  in  Pounds 


108.4 

I 
316.8 

410.8 

0 

Cent* 
Edge 

?r  of  panel  .  .  .  . 
of  side  belt  .  .  . 

.  .  .Observed 
.  .Computed 
.  .  .Observed 
.  .(  'omputed 

0.077 
0.130 
0.065 
0.093 

0.320 
0  .  380 
0.247 
0.271 

0.437 
0  .  500 
0.332 
0.357 

0 
0 

155 
124 

As  above  stated,  the  somewhat  large  excess  of  computed  over 
observed  deflections  is  due  to  two  circumstances:  first,  and  prin- 
cipally, to  the  age  and  consequent  stiffness  of  the  slab;  and  secondly, 
at  the  edge,  to  the  fact  of  a  single  panel  load  instead  of  many,  neither 
of  which  circumstances  is  taken  account  of  in  the  equations  used. 


236  COMPARATIVE    TEST    OF    TWO    SLABS 

3.     Comparative  Test  of  Norcross  and  Mushroom  Slabs.     This 

section  will  be  devoted  to  a  detailed  consideration  of  a  test  to  des- 
truction of  two  slabs,  12'  x  12'  between  column  centers,  con- 
structed for  experimental  purposes.  The  tests  were  made  by 
Professor  Win.  H.  Kavanaugh,  in  November  and  December, 
1912,  and  the  results  he  obtained,  together  with  a  mathematical 
discussion  based  upon  them,  will  be  here  given.  One  slab  was 
constructed  in  accordance  with  the  plans  and  specifications  of  the 
U.  S.  Patent  No.  698,542  issued  to  0.  W.  Norcross  for  a  slab  for 
flooring  of  buildings,  and  the  other  was  a  Turner  Mushroom  slab 
under  U.  S.  Patent  No.  1,003,384.  The  test  serves  to  bring  out  in 
a  striking  manner  not  only  how  two  slabs,  which  present  a  super- 
ficial resemblance  in  the  plan  of  arrangement  of  reinforcement, 
differ  from  an  experimental  and  practical  standpoint,  but  it  also 
makes  evident  their  radical  divergence  of  action  mechanically  and 
mathematically. 

That  two  slabs  of  the  same  span,  thickness  and  amount  of 
reinforcement  should  on  test  show  that  one  of  them  was  more  than 
twenty  times  as  stiff,  and  more  than  five  times  as  strong  as  the 
other,  and  that  the  failure  of  the  weaker  one  was  a  sudden  and 
complete  collapse,  with  little  or  no  warning  to  the  inexperienced 
eye,  while  the  other  gave  way  by  slowly  pulling  apart  little  by 
little,  thus  gradually  getting  out  of  shape  without  any  final  break 
down,  are  phenomena  that  deserve  the  close  attention  of  the  de- 
signer, and  are  of  the  highest  interest  scientifically  as  well  as  practi- 
cally. The  enormous  differences  in  the  deflections  and  in  the 
stresses  in  the  reinforcement  as  shown  by  extensomoter  measure- 
ments, and  in  the  character  of  the  failure  in  respect  of  safety  and 
its  relation  to  the  line  or  zone  of  weakest  section,  as  well  as  in  the 
difference  of  design  loads  and  breaking  loads  amounting  to  500%, 
all  illustrate  what  scientific  design  will  accomplish  and  what  results 
are  possible  by  an  ingenious  arrangement  of  the  reinforcement. 

These  slabs  were  each  of  the  same  thickness,  viz  6",  and  were  sup- 
ported by  columns  placed  at  the  corners  of  a  square  12'  x  12'  from 
center  to  center  of  columns.  The  slabs  projected  2'  to  3'  beyond 
the  centers  of  the  columns  on  each  side,  and  had  precisely  the  same 
number  and  size  of  reinforcing  rods  in  each  belt,  viz  eleven  3/8 
inch  round  rods.  The  concrete  was  of  a  1  :  2  :  4  mix,  and  while 
only  about  four  weeks  old  at  the  time  of  the  test,  it  had  been  poured 
warm  and  kept  warm  by  steam  heat  under  such  unusually  favorable 
conditions  as  to  have  become  well  cured  at  the  time  of  the  test. 
The  steel  used  showed  by  test  a  stress  at  yield  point  of  51,000  to 


BEAM    THEORY,    VERSUS    SLAB    THEORY  237 

55,000  pounds  per  square  inch,  and  an  ultimate  strength  of  76,000 
to  80,000  pounds,  with  an  elongation  of  twenty  to  twenty-five  per 
cent. 

The  first  slab  was  made  in  accordance  with  the  specifications 
of  the  Norcross  patent  already  referred  to  except  that  belts  of  rods 
were  substituted  for  the  netting  mentioned  by  the  patentee.  This 
design  was  selected  as  one  of  the  two  for  this  comparative  test, 
not  because  it  is  a  good  design,  or  one  that  any  engineer  would 
to-day  care  to  employ,  but  because  it  exhibits,  according  to  the 
express  intention  of  the  patentee,  simple  tension  on  its  lower  surface, 
everywhere  between  columns,  and  simple  compression  everywhere 
on  its  upper  surface  between  columns;  this  being  in  direct  contrast 
to  the  other  design,  which  is  arranged  not  only  to  resist  direct  ten- 
sions over  the  supports,  which  the  first  does  not,  but  also  to  resist 
circumferential  stresses  both  around  the  supports  and  around  the 
panel  centers,  as  any  truly  continuous  flat  slab  must. 

This  test  may  then  be  viewed  in  the  light  of  an  experimental 
demonstration  of  the  difference  between  a  reinforced  flat  slab  con- 
structed in  accordance  with  the  beam  theory  and  one  constructed 
in  accordance  with  correct  slab  theory,  where  true  and  apparent 
moments  differ  radically  as  shown  at  the  beginning  of  this  investi- 
gation, but  are  wholly  contradictory  to  any  form  of  simple  or  con- 
tinous  beam  theory.  This  test  may  be  regarded  as  settling  once  for 
all  the  question  of  applying  simple  beam  theory  to  a  cantilever  flat 
slab,  reinforced  throughout  practically  its  entire  area  with  a  lattice  of 
rods  crossing  each  other  and  in  contact.  It  shows  that  it  is  impos- 
sible to  compute  the  deflections  of  such  a  slab  by  beam  theory. 
Furthermore  this  impossibility  makes  it  certain  that  the  stresses 
in  such  a  slab  cannot  be  computed  by  beam  theory,  for  to  do  this  is 
to  commit  an  inconsistency  such  as  has  heretofore  too  often  been 
committed,  but  one  which  should  hereafter  be  carefully  avoided. 


238 


THE    NOKCROSS    TEST    SLAB 


Norcross  in  his  patent  already  referred  to  describes  his  con- 
struction as  consisting  "essentially,  of  a  panel  of  concrete  having 
metallic  network  encased  therein,  so  as  to  radiate  from  the  posts 

on  which  the  floor  rests The  posts  are  first  erected,  and  a 

temporary  staging  built  up  level  with  the  tops  of  posts.  Strips  of 
wire  netting  are  then  laid  loosely  in  place  on  top  of  the  staging .... 
The  concrete  is  then  spread  upon  or  moulded  in  place  on  the  staging 
to  enclose  the  metallic  network.  In  practice  I  have  sometimes 
laid  the  concrete  in  layers  of  different  quality,  the  lower  layer  of 
the  floor  which  encloses  the  wire  being  laid  with  the  best  concrete 

available If  the  forces  acting  upon  a  section  of  flooring 

supported  between  two  posts  be  analyzed  it  will  be  found  that  the 
tendency  of  the  floor  section  to  sag  between  its  supports  will  cause 
the  lower  layers  of  the  flooring  to  be  under  tension  while  the  upper 
layers  of  the  flooring  will  be  under  compression,  these  stresses  being, 
of  course,  the  greatest  at  the  top  and  bottom  layers,  respectively." 


Fig.  60.     Reinforcement  of  Xorcross  Slab 


THE    NORCROSS    TEST 


Fig.  61.     Norcross  Slab  Carrying  Load  3 


Col.  Cop  Plate  20*20*4 
Bel  fa  //-§  *  each 


Fig.  62.     Norcross  Slab 


240 


LOADS    ON    NORCROSS    SLAB 


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NORCROSS    SLAU  241 

The  number  and  arrangement  of  the  reinforcing  rods  in  the 
Norcross  experimental  slab,  (eleven  3/8"  round  rods  in  each  side 
and  diagonal  belt)  is  clearly  shown  in  the  view  of  Oct.  3 1st,  Fig.60,  which 
shows  the  forms  ready  for  pouring  the  concrete.  Steel  plates 
20"  x  20"  x  0.5"  carry  the  rods  and  rest  on  the  tops  of  the  columns, 
which  last  in  this  case  consisted  of  steel  pipes  about  5J"  in  dia- 
meter filled  with  concrete  and  embedded  at  their  lower  ends  in  large 
concrete  blocks.  A  vertical  central  bolt  in  the  concrete  at  the 
upper  end  of  each  pipe  permitted  the  plates  to  be  firmly  secured  to 
the  tops  of  the  columns.  The  view  of  Nov.  30th,  Fig.  61,  clearly 
shows  the  manner  of  placing  the  pig  iron  on  the  slab  for  load  3. 
This  slab  is  16'  x  16'.  The  loading  at  first  covered  an  area  having 
the  form  of  a  Greek  cross  whose  central  square  was  five  feet  on  a 
side  with  arms  5'  6"  long,  as  represented  in  accompanying  diagram 
of  loaded  areas  A,  B,  C,  D,  E,  Fig.  62,  and  of  amounts  shown  in 
Table  3. 


Fig.  63.     Collapse  of  Norcross  Slab 

When  10,000  pounds  had  been  piled  on  the  central  part  of  the 
slab  in  addition  to  load  No.  4,  of  66,812  pounds,  the  slab  suddenly 
failed.  In  anticipation  of  such  failure  timber  blocking  had  been 
placed  under  the  slab  to  prevent  its  falling  more  than  possibly  ten 
or  twelve  inches. 


242 


NORCROSS    TEST 


Fig.  64.     Collapse  of  Nortross  Slab 

The  two  views  of  Dec.  2d,  Fig.  63  and  Fig.  64,  show  the  con- 
dition of  the  slab  after  removing  part  of  the  final  loading  in  order 
to  render  the  nature  of  the  failure  visible.  Careful  extensometer 
measurements  of  the  elongations  of  the  steel  rods  at  the  middle 
of  the  side  and  diagonal  belts  were  made  under  the  action  of  loads 
1,  2,  3  and  4,  and  also  similar  extensometer  measurements  in  the 
concrete  both  on  the  top  and  the  bottom  of  the  slab  along  the  center 
line  of  the  side  and  diagonal  belts  near  those  edges  of  two  of  the 
steel  plates  which  were  nearest  the  center  of  the  belts.  Besides 
these,  certain  other  measurements  of  the  concrete  were  made  at 
right  angles  to  the  diagonals.  Deflections  were  also  measured 
under  these  loads  at  the  middle  of  the  diagonal  belt  and  of  two  of 
the  side  belts  at  V,  W,  X,  Y,  Z. 

These  measurements  all  show  beyond  question  that  the  side 
and  diagonal  belts  act  like  simple  beams  in  this  form  of  construction, 
since  the  stresses  in  the  steel  and  concrete  on  the  under  side  of 
the  slab  in  the  direction  of  the  rods  is  invariably  tensile,  while  the 
stresses  in  the  same  directions  on  top  of  the  slab  are  always  com- 
pressive.  It  was  the  avowed  intention  of  Norcross  to  reinforce 
the  slab  in  this  manner  since  he  regarded  the  upper  part  of  the  slab 
as  being  subjected  everywhere  to  compression  and  the  lower  part 
to  tension  only,  as  stated  in  his  specifications  as  already  quoted. 


COMPUTATION    OF    THE    NORCROSS    TEST  243 

The  following  computation,  Table  4,  shows  a  good  approxi- 
mate agreement  of  the  results  of  this  test  with  the  beam  theory  of 
flexure,  assuming  for  simplicity  that  the  stiff  steel  supporting  plate 
and  interlacing  of  the  ends  of  the  belts  diminishes  the  effective 
span  of  the  side  belts  by  12",  and  the  diagonals  in  the  same  pro- 
portion, and  further  assuming  that  the  loading  was  all  applied  at 
the  middle  of  the  side  and  diagonal  belts. 

The  extensometer  measurements  made  were  for  a  length  of 
8",  consequently  the  stress  in  the  steel  per  square  inch  would  be 
computed  thus: 

/s  =  l/8   (elongation  in  8")  x  30,000,000; (IJj 

and,  this  being  known  from  observation,  it  will  be  possible  to  com- 
pute the  load  W  carried  by  the  beam  in  which  the  given  elongation 
occurs,  as  follows : 

The  bending  moment  due  to  a  concentrated  load  W  at  the  mid- 
dle of  a  beam  of  length  L  is  M  =  \  W  L, (2)1 

and  the  equal  moment  of  resistance  of  the  reinforcement  by  which 

it  is  held  in  equilibrium  is    M  =  A  j  d  fs (3)i 

in  which  A  is  the  total  cross  section  of  the  steel  in  the  belt  = 
11  x  0.11  =  1.215  sq.  in.,  and  the  distance  from  the  center  of 
the  steel  to  the  center  of  compressive  resistance  of  the  concrete 
is  assumed  to  be,  j  d  =  0.9  x  5.75 

when  d  =  5.75  is  taken  as  the  distance  from  the  center  of  action 
of  the  steel  to  the  top  of  the  slab, 

Hence  W   =   4  A  j  d  fs/L (4)i 

is  the  load  required  to  cause  the  stress  /8  in  the  steel.  In  the  side 
belts  we  assume  the  span  L  to  be  132",  and  in  the  diagonals  132  V2. 

In  Table  4,  which  follows,  it  will  be  noticed  that  loading  No.  1 
is  too  small  to  develop  sufficient  elongations  or  deflections  to 
overcome  the  initial  compressions  in  the  concrete  in  which  the 
reinforcement  is  embedded,  so  that  the  load  carried  by  the  steel  is 
only  about  one  half  of  the  actual  load,  the  other  half  being  evidently 
carried  by  the  concrete  in  which  it  is  embedded.  This  is  in  com- 
plete accord  with  other  similar  experiments.  But  in  case  of  loads 
No.  2  and  No.  3,  where  the  steel  is  stressed  close  to  the  yield 
point,  the  sum  of  the  loads  as  shown  by  the  stresses  in  the  steel 
is  very  close  to  the  total  actual  load.  It  is  assumed  that  these 
total  actual  loads  are  carried  by  the  various  belts  in  the  same  pro- 
portion as  the  computed  loads,  since  there  is  no  other  way  of 
dividing  the  total  load  between  the  belts.  This  may  be  stated 
mathematically,  as  follows: 


244 


LOADS    AND    DEFLECTIONS    OF    NOROROSS    SLAB 


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NORCROSS  SLAB  ON  THE  BEAM  THEORY  24") 

Let  Wi  =  the  computed  load  on  a  side  belt, 
and  W2  =  the  computed  load  on  a  diagonal  belt. 
Let  W\   =  the  actual  load  on  a  side  belt, 
and  W2  =  the  actual  load  on  a  diagonal  belt. 
Then  4Wi  +  2W2  =  total  computed  load  on  slab. 
and  4Wi  +  2W2  =  total  actual  load  on  slab. 

4  W i  +  2  W2        TFi'        T^a 

Then  -  - (5)i 

4W1  +  2W2        W1        W2 

from  which  W[  and  TF 2  can  be  computed,  Wi  ,  T72  and  4TFi  +  4TF 2 
being  already  known. 

The  distribution  of  load  No.  4  is  not  such  as  to  render  this 
method  of  computing  it  so  applicable  as  to  other  loads.  Having 
found  the  actual  distribution  of  loading  W[  and  W2  the  center 
deflections  of  the  belts  have  been  computed  by  simple  beam  theory 
from  the  formula. 

TF'L3 

D2  =   -  - (6)i 

48  E  A  i  j  d2 

in  which  i  d  =  the  distance  from  the  steel  to  the  neutral  axis  and 
the  value  of  i  has  been  assumed  to  be  0.69;  W'  is  the  actual  load  on 
the  belt  and  L  is  its  span  as  previously  stated. 

It  appears  from  Table  4,  that  the  effect  of  the  reinforcement 
is  accounted  for  to  a  reasonably  close  approximation  by  consider- 
ing the  belts  to  act  as  a  combination  of  simple  beams,  at  least  with- 
in the  range  of  loading  near  the  yield  point  of  the  steel. 

It  appears  that  the  steel  reached  its  yield  point  under  a  total 
load  on  the  slab  of  from  15  to  18  tons  and  final  collapse  occured  under 
a  total  load  of  a  little  over  twice  the  latter  amount  not  distributed 
uniformly  but  piled  more  in  the  general  form  of  a  pyramid. 

It  was  observed  that  the  application  of  the  relatively  small 
loading  on  the  corner  areas  F,  G,  H,  I,  had  a  very  injurious  effect 
upon  the  slab,  tending  to  break  it  across  the  tops  of  the  columns. 

The  results  of  the  test  may  be  summarized  in  the  Norcross 
system  as  follows: 

1st.  This  slab  is  of  the  simple  beam  type,  and  the  test  shows 
no  cantilever  action  and  no  circumferential  slab  action. 

2nd.  The  narrow  belts  running  diagonally  leave  large  areas 
without  reinforcement,  and  there  is  consequently  no  provision  for 
resisting  circumferential  tensions  as  required  in  slab  action. 

3rd.  The  concrete  showed  compressive  stresses  on  the  upper 
surface  of  the  slab  in  the  direction  of  all  the  reinforcing  rods. 


24b  SUMMARY    OF    NORCROSS    TKST 

4th.  The  concrete  showed  tension  at  the  bottom  surface  in 
the  direction  of  all  the  reinforcing  rods,  in  agreement  with  Norcross' 
own  analysis. 

5th.  This  slab  deflected  1.6"  under  33  tons  and  then  broke 
down  completely  under  38  tons. 

6th.  The  first  crack  appeared  under  a  load  of  15  tons  and 
deflection  of  0.7". 

7th.  The  slab,  not  being  reinforced  on  the  top  surface  over 
the  columns,  inevitably  cracks  at  a  column  when  the  slab  is  loaded 
around  the  column. 

8th.  At  failure  the  steel  had  passed  its  yield  point.  The 
percentage  of  reinforcement  in  the  diagonal  belt  if  we  regard  the 
belt  as  about  18"  wide  is  very  nearly  1%,  but  since  a  width  of 
concrete  somewhat  greater  than  that  may  be  assumed  to  act  with 
this  steel,  the  percentage  of  reinforcement  is  somewhat  less  than 
1%.  Similarily,  the  side  belts  of  width  36"  have  a  reinforcement 
less  than  0.5%.  The  full  strength  of  the  steel  in  both  belts  was 
developed  by  the  concrete,  which  fact  demonstrates  that  the  con- 
crete was  of  high  grade  and  well  cured.  The  steel  was  also  of 
good  standard  quality,  and  the  test  was  therefore  in  every  way 
fair  to  the  Norcross  slab,  since  it  was  so  loaded  as  to  cause  the 
stresses  in  the  side  and  diagonal  belts  to  be  practically  equal,  thus 
using  the  steel  most  economically.  The  slab  failed  because  the 
steel  yielded  near  the  middle  of  the  spans,  thus  causing  the  concrete 
above  the  steel  to  crack  and  break. 

The  second  slab  was  made  according  to  the  Turner  Mush- 
room System,  under  the  patent  already  referred  to. 

Since  all  forces  in  a  plane  may  be  resolved  into  components 
along  any  pair  of  axes  at  right  angles  to  each  other  it  is  possible 
to  provide  reinforcement  to  resist  any  horizontal  tensile  stresses 
in  the  slab  by  various  arrangements  of  intersecting  belts  of  rods  at 
zones  where  these  stresses  occur.  The  combination  of  such  belts  with 
radial  and  ring  rods  to  constitute  a  large  and  substantial  canti- 
lever mushroom  head  at  the  top  of  each  column  affords  a  very 
effective  and  economical  arrangement  for  controlling  the  distribution 
of  the  stresses  in  the  slab,  and  it  places  the  reinforcement  where 
it  is  most  needed.  It  not  only  has  the  same  kind  of  advantage 
that  the  continuous  cantilever  beam  has  over  the  simple  girder 
for  long  spans,  but  combines  with  it  the  kind  of  superiority  that  the 
dome  has  over  the  simple  arch  by  reason  of  circumferential  stresses 
called  into  play,  which  greatly  adds  to  the  carrying  capacity  of  the 
slab. 


REINFORCEMENT    OF    MUSHROOM    TEST    SLAB 


247 


Fig.  65.     Reinforcement  of  Mushroom  Slab 


Column   RoJS   8-f&  + 

Fig.  66.     Mushroom 


Diom. 


248 


THE    MUSHROOM    SLAB    TEST 


The  mushroom  test  slab  was  six  inches  thick,  and  was  sup- 
ported on  four  IS"  by  18"  square  reinforced  concrete  columns 
distance  12'  from  center  to  center.  These  had  square  capitals, 
42"  x  42".  The  slab  was  appromimately  18'  x  18',  and  the  dia- 
meter of  the  outer  ring  rod  of  the  Mushroom  was  66",  while  the 
inner  ring  was  42".  These  were  supported  on  eight  1-1/8"  round 
radial  column  rods. 


Fig.  67.     Mushroom  Slab,  Load  4. 

This  will  be  clearly  understood  from  the  view  dated  October 
31st,  Fig.  65,  which  shows  the  reinforcement  and  forms  ready  for 
pouring  the  concrete.  The  remaining  views  are  explained  by  their 
accompanying  legends. 

The  diagram  of  loaded  areas  for  the  mushroom  slab  Fig.  66,  is 
like  that  already  given  for  the  Norcross  slab  in  every  particular 
except  that  the  size  of  the  mushroom  slab  being  18'  x  18',  while  the 
Norcross  slab  was  16'  x  16',  the  arms  of  the  Greek  cross  in  the 
mushroom  slab  are  each  6'  6"  long  and  5'  wide. 


LOADS    ON    MUSHROOM    SLAB 


249 


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LOADS    ON    MTSHROOM    SLAB 


251 


Fig.  68.     Mushroom  Slab,  Load  7. 


Fig.  69.     Mushroom  Slab,  Load 


252  COMPUTATION    OF    MUSHROOM    TEST 

The  accompanying  Table  5,  exhibits  the  loads  per  square  foot 
of  each  of  the  subsidiary  areas  shown  in  the  diagram  as  also  the 
total  loads  on  each  of  those  areas.  The  view  of  Dec.  3,  Fig.  67, 
shows  load  4,  and  that  of  Dec.  13,  Fig.  68,  load  7,  while  that  of 
Dec.  16,  Fig.  69,  shows  load  9. 

Elongations  of  steel  were  measured  by  Berry  extensometers 
in  two  of  the  side  belts  and  in  one  of  the  diagonal  belts  until  the 
concrete  began  to  fail  under  loads  Nos.  7  and  8.  Deflections 
were  also  measured.  In  Table  6,  these  will  be  considered  so  far  as 
they  relate  to  the  middle  points  of  the  belts.  Loads  8,  9,  10,  are 
of  great  interest  as  exhibiting  the  behavior  of  the  slab  under  ex- 
cessive loads,  showing,  as  they  do,  yielding  and  large  permanent 
deformation  without  dangerous  collapse. 

By  (52)  the  uniformly  distributed  load  per  square  foot  of 
panel  area  when  the  stress  in  the  diagonal  belt  is  fa  is  found  for  a 
square  panel  from  the  expression 


256  j  d2  A 
144  L 


=   w  =  TF/144  =  -  —  /8  ...............  ...  (52a) 


which  applied  to  this  slab  gives  us 

256  X  0.89  X  5.125  X  1.215 

w    =   -  -/.  =  A/14.6  ........  (52b) 

144  X  144 

The  values  of  this  uniformily  distributed  load  w  is  tabulated 
in  table  4,  for  each  of  the  observed  values  of  the  /„  in  the  diagonal 
belts.  The  values  of  w  so  computed  tend  to  become  identical, 
in  case  of  the  heavier  loads,  with  the  loads  per  square  foot  on  the 
central  area  C,  as  might  reasonably  be  expected,  w  being  the  uniformly 
distributed  load  which  is  equivalent  so  far  as  the  stress  on  the  dia- 
gonal belt  is  concerned  to  the  action  of  the  actual  loads  which  are 
not  uniformly  distributed. 

Now  compute  by  (54),  (55),  (58),  (60)  and  (61),  the  deflections 
at  the  mid  side  belt  and  at  center  of  the  panel,  due  to  a  uniform  load. 
These  results  are  given  in  Table  4,  and  accord  closely  with  those 
actually  observed,  as  they  should,  because  the  irregularity  of  dis- 
tribution does  not  produce  deflections  that  differ  much  from  the 
equivalent  uniform  load  as  computed  above. 

In  these  computations  it  is  assumed  that  di  =  5.5",  d2  = 
5.125",  d  =  4" 


DEFLECTIONS    OF    MUSHROOM    SLAB 


253 


The  double  set  of  values  under  loads  4  and  5  is  due  to  the 
fact  that  readings  were  had  under  load  4,  immediately  after  the 
load  was  applied,  and  again  7  days  later  before  applying  load  5. 
The  second  set  of  readings  were  the  larger  as  shown.  The  second 
set  of  readings  under  load  5,  were  taken  four  days  subsequently 
to  the  first  set. 

It  appears  from  Table  6,  that  the  observed  results  are  account- 
ed for  by  the  slab  theory  to  a  good  degree  of  approximation 
so  long  as  the  concrete  was  intact. 


Seers?.' <?/7<  scua-e  eo> /&/*>  o.je  tenth  /nch 


Fig.  70.     Comparative  Deflections  of  Norcross  and  Mushroom  Slabs. 


A  graphical  representation  of  the  experimental  observations 
in  the  deflections  at  the  points  V,  W,  X,  Y,  Z,  of  the  two  slabs  is 
found  in  Fig.  70,  which  shows  in  a  striking  manner  how  small  the 
loads  and  how  great  the  deflections  were  in  the  Norcross  slab  on  the 
one  hand,  and  how  large  the  loads  and  how  small  the  deflections 
were  in  the  mushroom  slab  on  the  other  hand. 


254 


LOADS    AND    DEFLECTION'S    OF    MUSHROOM    SLAB 


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DISCUSSION    OF    DEFLECTIONS    OF    THE    TWO    SLABS  255 

It  will  be  seen  from  Tables  3  and  5,  that  the  first  three  loads 
were  practically  the  same  for  both  slabs.  In  the  Norcross  slab 
load  3,  of  18  tons,  stressed  the  steel  up  to  the  yield  point,  but  in 
the  mushroom  slab  the  stress  was  so  small,  (being  in  fact  less  than 
ten  per  cent  of  the  former)  as  probably  not  to  remove  all  the  com- 
pression from  the  concrete  in  which  it  was  embedded.  Indeed  the 
load  on  the  latter  slab  became  five  times  as  much,  90  tons,  without 
stressing  its  steel  out  to  the  yield  point,  at  which  time  it  was  carry- 
ing about  twice  the  load  which  caused  the  complete  failure  of 
the  Norcross  slab. 

Moreover  the  deflection  of  the  Norcross  slab  under  load 
3,  was  twenty-two  times  that  of  the  mushroom  slab  under  the 
same  load.  This  result  is  in  full  accord  with  slab  theory  which  shows 
that  the  central  deflection  of  a  continuous  diagonal  beam  with  fixed 
ends  uniformly  loaded  with  one  sixth  of  the  total  load  on  the  slab 
and  having  the  same  thickness  and  reinforcement  as  the  diagonal 
belt,  would  have  more  than  six  times  the  central  deflection  of  the 
slab,  while  the  stress  in  its  steel  would  be  three  or  four  times  as 
much.  This  gives  a  measure  of  the  effect  of  slab  action. 

By  the  phrase  "slab  action"  we  designate  the  increased  strength 
and  stiffness  of  the  slab  by  reason  of  its  resistance  to  circumferential 
stresses  around  the  columns  and  around  the  center  of  the  panel. 

Furthermore,  if  this  continuous  beam  be  compared  with  a  simple 
beam  uniformly  loaded  and  having  the  same  reinforcement,  the 
latter  would  have  five  times  the  deflection  of  the  continuous  beam, 
or  thirty  times  that  of  the  slab,  while  the  stress  in  the  steel  would 
be  one  and  one-half  times  that  in  the  continuous  beam,  and  six  or 
seven  times  that  in  the  slab.  This  last  exhibits  the  effect  of  canti- 
lever action  combined  with  slab  action. 

The  apparent  discrepancy  between  the  observed  ratio  of  de- 
flections in  these  two  slabs  of  22  and  the  just  computed  deflections 
of  30,  is  to  be  accounted  for  by  the  fact  that  the  computation 
assumed  equal  spans,  whereas  the  Norcross  span  was  assumed 
to  be  diminished  from  144"  to  132"  by  the  column  plate.  A  re- 
duction of  the  span  of  this  amount  will  change  the  computed  de- 
flections in  the  ratio  of  1443  :  1323  :  :  30  :  23  which  is  in  practical 
agreement  with  the  observed  result  of  22. 


256  SUMMARY    OF    TEST    OF    MUSHROOM    SLAB 

By  the  phrase  "cantilever  action"  we  designate  the  increased 
strength  and  stiffness  which  is  due  to  the  continuity  of  the  beam 
or  slab  at  its  supports  so  that  it  is  convex  upwards  at  such  points. 

While  the  concentration  of  the  loading  toward  the  middle  of 
the  panel,  such  as  was  the  case  in  this  test,  may  prevent  any  pre- 
cise agreement  of  these  numerical  estimates  based  on  uniform 
loading  with  the  results  of  the  tests,  they  cause  the  general  agree- 
ment shown  in  the  tables  and  tend  strongly  to  sustain  our  confi- 
dence in  the  validity  of  the  analysis  from  which  these  concordant 
approximate  estimates  are  obtained. 

The  amazing  difference  in  the  strength  and  stiffness  of  these 
two  slabs,  which  contain  practically  the  same  amount  of  concrete 
and  steel,  is  due  to  the  difference  of  principle  of  their  construction, 
which  may  be  summarized  for  the  mushroom  system  by  consider- 
ing its  slab  action  and  its  cantilever  action  under  the  following 
counts,  viz: 

1st.  Circumferential  slab  stresses  around  the  column  heads  are 
most  economically  and  effectively  provided  for  by  the  ring  rods  and 
the  octagonal  interlacing  of  the  slab  rods. 

2nd.  The  size  of  the  mushroom  heads  is  such  as  to  make  the 
belts  so  wide  as  to  provide  reinforcement  over  the  entire  area  of 
the  slab,  thus  securing  slab  action  in  the  central  part  of  the  panel 
where  the  belts  lie  near  the  lower  surface. 

3rd.  The  reinforcing  belts  cover  a  wide  zone  at  the  top  of 
the  slab  over  the  columns  and  mushroom  head,  which  thus  provides 
resistance  to  tension,  and  ensures  effective  cantilever  and  slab  action. 

4th.  Concrete  is  thus  stressed  in  compression  at  the  bottom 
of  the  slab  for  a  wide  zone  around  the  columns. 

5th.  Under  a  load  equal  to  the  breaking  load  of  the  Norcross 
slab,  amounting  to  thirty-eight  tons,  the  mushroom  slab  deflected 
at  first  only  1/8",  but  after  exposure  to  rain  and  great  changes  of 
temperature  for  seven  days  had  somewhat  softened  the  concrete 
the  deflection  increased  to  1/4". 

6th.  The  first  crack  appeared  underneath  the  edge  of  the 
slab  across  the  side  belt  under  load  No.  5,  of  fifty-six  tons,  with  a 
center  deflection  of  0.4"  and  an  average  deflection  at  the  middle 
of  side  belts  of  0.25". 

7th.  No  cracks  appeared  on  the  upper  side  of  slab  at  the 
edge,  nor  were  any  seen  elsewhere,  until  load  No.  7,  of  90  tons  was 
applied,  when  a  center  deflection  of  1/2"  was  reached. 


FAILURE    OF    MUSHROOM    SLAB 


257 


Fig.  71.     Failure  of  Mushroom  Slab. 


___!_       BBBHMPBI 

Fig.  72.     Failure  of  Mushroom  Slab.     Load  Removed. 


258  FAILURE    OF    MUSHROOM   TEST   SLAB 

8th.  The  slab  carried  its  final  load  of  over  120  tons  for  twenty- 
four  hours  without  giving  way.  It  demonstrated  the  impossibility 
of  its  sudden  failure  by  gradually  yielding  until  it  reached  a  final 
deflection  of  some  nine  inches,  as  seen  in  the  views  of  Dec.  17th 
and  24th,  Figs.  71  and  72. 

9th.  While  the  slab  steel  in  each  belt  was  the  same  as  in  the 
Norcross  slab,  the  crossing  of  the  belts  increased  the  percentage 
of  slab  reinforcement  so  much  above  that  of  the  simple  belt  rein- 
forcement that  stress  in  the  steel  did  not  pass  the  yield  point  and 
the  failure  was  largely  due  to  the  giving  way  of  the  concrete  around 
the  cap,  but  partly  to  some  yielding  at  the  line  of  weakest  ultimate 
resistance,  both  of  which  statements  are  confirmed  by  the  view  of 
Dec.  24th,  Fig.  72,  where  the  removal  of  the  loading  permits  the 
irregular  circular  line  previously  mentioned  to  be  made  out  at  a 
distance  from  the  center  of  each  column  of  somewhat  less  than  L/2. 

Less  steel  is  required  in  this  system  than  in  the  Nor- 
cross slab  for  the  same  limiting  stresses.  Since  the  steel  in  this 
slab  did  not  pass  the  yield  point  any  greater  percentage  of  reinforce- 
ment would  be  useless  and  would  not  increase  the  strength  of  the 
slab.  It  has  been  found  that  good  practice  requires  a  percentage 
of  steel  dependent  in  the  following  manner  upon  the  thickness 
of  the  slab: 

lid  =  L/35  the  belt  reinforcement  =  0.2% 
lid  =  L/24  the  belt  reinforcement  =  0.3% 
lid  =  L/20  the  belt  reinforcement  =  0.4% 

Comparision  of  the  steel  in  the  test  slabs:  Norcross.  Mushroom. 

Size  of  slab 16'  x  16'  18.4'  x   17.8' 

Area  of  slab 256  sq.  ft.  328  sq.  ft. 

Length  of  3/8"  rods  in  the  slab 1188  ft.  1750  ft. 

Weight  of  3/8"  rods  in  the  slab 446  Ibs.  650  Ibs. 

Weight  of  Plates  or  Heads  in  the  slab. .  .  228  Ibs.  435  Ibs. 

Total  weight  of  steel  in  the  slab 674  Ibs.  1085  Ibs. 

Weight  of  steel  per  square  foot  of  slab..  2.6  Ibs.  3.3  Ibs. 

Area  of  Panel  12  x  12  ft 144  sq.  ft.  144  sq.  ft. 

Length  of  slab  rods  per  panel 638  ft.  638  ft. 

Weight  of  slab  rods  per  panel 239  Ibs.  239  Ibs. 

Weight  in  plates  or  heads  per  panel 57  Ibs.  109  Ibs. 

Total  weight  of  steel  per  panel 296  Ibs.         348  Ibs. 

Weight  of  steel  per  square  foot  of  panel.        2.06  Ibs.  2  5/12  Ibs. 


STRESS    AND    DEFLECTION    RATIOS 


259 


The  results  of  this  comparative  test  of  two  slabs  lead  to  several 
interesting  applications  of  the  principles  previously  cited  under  the 
treatment  of  such  slabs  as  a  kind  of  mechanism  amenable  to 
analysis  by  the  theory  of  work. 

The  law  of  conservation  of  energy  teaches  that  for  elastic  de- 
flections the  average  stress  for  instance  at  mid  span  of  a  floor  slab 
of  one  kind  compared  with  the  average  stress  at  mid  span  of  another 
kind  of  the  same  uniform  thickness  and  percentage  of  steel  at  the 
sections  compared,  will  bear  the  same  ratio  as  the  respective  deflec- 
tions under  identical  loads  producing  these  stresses.  In  other  words, 
a  radical  difference  in  mode  of  action  brought  about  by  a  different 
arrangement  of  the  reinforcement  in  the  slabs  will  not  affect  the 
validity  of  this  deduction.  The  experimental  data  supplied  by  the 
two  respective  slabs,  the  Norcross  and  the  Mushroom  type  slab,  is 
of  interest  in  this  connection. 

Compare  the  loads  which  were  applied  in  Greek  cross  form, 
the  areas  A,  B,  D,  E,  loaded  alike,  but  the  area  C  loaded  at- 
tunes, with  approximately  double  the  load  on  the  arms  of  the  cross 
—the  idea  being  to  throw  upon  the  respective  belts  crossing  the  area 
C  approximately  the  same  load  as  on  each  direct  belt  as  shown  in 
the  following  table: 

TABLE   7. 


Load 

Steel 

Stress 

On    each 

Average 

Type 

Area     On   Area 

Total 

on    belts 

Ratio 

De- 

Ratio 

ABED 

C 

at  mid 

flection 

span 

Norcross  

3138 

2852 

15404 

7690 

.2207 

Mushroom.  .  .  . 

3276 

2532 

15624 

643 

12 

.017 

12.8 

Norcross  

6288 

6002 

31154 

27133 

.707 

Mushroom  .... 

6552 

5040 

31248 

1154     !     23.5 

.0353 

20. 

Norcross.  .  . 

6288 

11420 

36572 

34894 

1.020 

Mushroom  .... 

6532 

10458         36666 

1472 

23.4 

.0463 

22. 

The  equality  of  the  ratios  above  compared  for  the  respective 
steel  stresses  and  deflections  is  in  most  satisfactory  agreement  with 
the  theory  when  the  lack  of  perfect  identity  of  application  of  the  load 
as  regards  quantitive  distribution  and  the  failure  of  the  observers 
to  measure  stresses  and  deflections  simultaneouslv  is  considered. 


It  will  be  noted  that  these  ratios  remain  substantially  the  same, 
as  nearly  as  can  be  expected  considering  the  fact  that  the  measure- 


260  RATIOS    OF    STRESS    IN    DIAGONAL    AND    SIDE    BELTS 

ments  of  the  deflections  and  the  measurements  of  the  steel  stresses 
were  not  simultaneous,  there  being,  however,  some  time  effect  as 
indicated  by  the  diagrams. 

These  tests  show  a  surprising  difference  in  stiffness  for  the  same 
cross  section  of  steel  in  diagonal  belts,  and  substantially  the  same 
difference  so  far  as  its  efficiency  is  concerned  in  direct  belts.  The 
enormous  difference  in  stiffness — the  one  being  22  times  as  stiff  as 
the  other — is  a  difference  little  short  of  amazing,  when  we  consider 
the  fact  that  a  continuous  beam  is  only  five  times  as  stiff  as  a  simple 
beam.  Hence  the  test  shows  clearly  that  the  mode  of  operation 
must  involve  a  far  more  radical  difference  than  mere  continuity  of 
reinforcement. 

The  difference  that  is  most  striking  in  the  arrangement  of  rein- 
forcement in  these  two  slabs,  lies  in  the  difference  in  the  width  of 
the  belts,  the  material  in  the  Norcross  type  slab  being  concentrated 
in  narrow  strips  diagonally,  while  the  material  in  the  Mushroom 
type  is  spread  out  from  three  to  four  times  as  wide,  and  covers 
the  area  of  the  slab  fully. 

A  comparison,  then,  of  the  average  stress  in  the  respective  belts 
at  mid  span,  should  bring  out  in  a  striking  manner,  the  difference 
in  mode  of  action,  because  of  the  difference  in  the  width  of  the  belts 
in  the  two  areas.  This  comparison  should  show  the  difference  in 
effect  at  mid  span  of  spreading  the  belts  as  against  concentrating  the 
metal  in  relatively  narrow  diagonal  beam  strips.  The  following 
table  of  average  stresses  is  submitted  in  order  to  show  the  ratio  of 
the  stress  in  the  diagonal  belt  compared  to  that  in  the  direct  belt : 

NORCROSS  SLAB 


Load  Stresses  in  Direct  Belts  Stresses  in  Diag-  Ratio   diag.    to   direct   belt.     Stress 
in  Ibs.  per  sq.  inch.          onal  Belts  average. 


3    34,650    34,690     35,213  1.02 


4    42,800    42,050     45,880 


1.08 


MUSHROOM  SLAB 


Load   Stresses  in  Direct  Belts  Stresses  in  Diag-  Ratio  diag.   to  direct   Belt.    Stress 
in  Ibs.  per  sq.  inch.  onal  Belts  Average 


4      9,677 
7     25,380 


12,627     3,977 
31,490     13,713 


.36 

.484 


The  ratio  of  the  diagonal  belt  stress  to  the  direct  belt  stress  is 
what  was  to  be  expected  in  the  Norcross  type  slab  on  the  beam  strip 
theory;  and  this  is  to  be  accounted  for  by  the  unreinforced  triangular 
areas  in  the  arrangement  of  the  metal. 


RELATIVE  INCREASE  OF  STRESSES  AND  DEFLECTIONS  WITH  LOADING     261 

The  spreading  out  of  the  belt  reverses  the  ratio  found  in  the 
Mushroom  slab  and  changes  it  so  that  the  relative  stress  in  the  diago- 
nal belt  at  mid  span  of  the  Norcross  slab,  when  compared  with  the 
stress  in  the  direct  belts  is  one  to  two  hundred  percent  greater  than 
the  similar  ratio  in  the  case  of  the  Mushroom  slab;  and  this  striking 
difference  between  the  two,  must  account  then,  in  a  large  part,  for 
the  amazing  difference  in  deportment  of  the  two  respective  tests 
slabs.  It  should  be  noted  that  the  Mushroom  test  slab  was  at  a 
great  disadvantage  on  account  of  the  Greek  cross  type  of  loading 
suggested  by  Mr.  Eddy,  which  was  very  advantageous  to  the  Nor- 
cross type  slab,  since  it  would  cause  the  greatest  stress  at  that  sec- 
tion of  the  slab  best  able  to  resist  the  same;  while  in  the  case  of  the 
Mushroom  slab,  a  uniform  load  over  the  column  areas  would  produce 
cantilever  action  which  would  tend  to  neutralize  the  stress  at  mid 
span. 

Refer  now  to  the  diagram  in  Fig.  70,  page  253,  where  the  deflec- 
tions of  the  Mushroom  and  Norcross  test  slabs  have  been  plotted. 

The  Norcross  type  slab  with  the  steel  at  the  bottom  is  known  by 
the  application  of  the  law  of  rigidities  to  exhibit  in  its  action  the 
predominant  phenomena  and  characteristics  of  the  simple  beam. 
If  this  deduction  be  correct,  a  rapid  rise  in  steel  stress  should  be  shown 
by  the  yielding  of  the  matrix  under  the  larger  loads  thru  over-strain 
by  the  action  of  indirect  tensions  combined  with  direct  tensions. 
The  following  table  is  instructive  in  regard  to  these  ratios. 

NORCROSS  TEST  SLAB 


Load  in  Terms  of 

Load  1. 

Steel  Stresses  in  Ratios 
of    that   for     Load    1. 

Deflections  in  Ratios  of 
that  for  Load  1 

Load  1 

7690 

.2207 

Load  2  —  Load  1  x 

2.01  

7690  x  3  .  54 

.2207  x3.21 

Load  3  —  Load  1  x 

2.38  

7690  x4.53 

.2207  x4.65 

Observe  from  this  table  the  leaking  out  of  the  potential  energy 
stored  in  the  concrete,  as  shown  by  the  increase  in  the  steel  stress 
by  two  hundred  and  fifty  percent  for  a  one  hundred  percent  increase 
in  load;  or  under  Load  3,  an  increase  of  three  hundred  and  fifty 
percent  for  an  increase  in  load  of  one  hundred  and  forty  percent. 

Now  compare  this  with  the  deportment  of  a  true  multiple-way 
reinforced  slab,  which,  by  the  law  of  rigidities  must  act  as  a  true 
continuous  plate,  instead  of  an  aggregation  of  simple  beam  strips. 


262        RELATIVE    INCREASE    OF    STRESSES    AND    DEFLECTIONS    WITH    LOADING 

If  the  preceding  theory  of  storage  of  potential  energy  be  correct, 
the  storage  reservoir  of  indirect  tensile  energy  found  in  this  case 
should  be  dependable  or  without  leak  in  the  slab  and  a  uniform  in- 
crease in  stress  and  deflection  with  increase  in  loads  should  be  the 
phenomena  found. 

MUSHROOM  TEST  SLAB 


Load  in  terms  of 
Load=  15624 

Average  Stee)  Stress  a't 
mid  span  in  terms 
of  that  for  Load  1  . 

Deflection  in  terms 
that  for  Load  1 

of 

Load  1  —  15624 

643 

017 

Load  2  =  15624  x 

2 

643  x  1  8 

017  x  2  07 

Load  3  =  15621  x 
Load  4  =15624  x 

2.38  
4.9    

643  x2.3 
643  x  6  .  34 

.017  x2.7 
.017  x7.12 

Loads  1,  2,  and  3,  on  the  Norcross  test  slab,  and  loads  1,  2,  3, 
and  4,  on  the  Mushroom  test  slab,  are  strictly  comparable,  since  for 
these  loads,  there  was  identity  in  location,  and  approximate  equality 
in  the  time  element  of  loading. 

The  proportionality  of  increase  of  stress  and  deflection  with  the 
load  in  this  table  of  the  Mushroom  test,  contrasts  strongly  with  the 
absence  of  such  proportionality  in  the  Norcross  test  slab  deport- 
ment. Each  test  slab  was  supported  upon  columns,  (see  Figures 
63  and  67)  and  the  stress  and  deflection  in  each  test  is  affected  to  some 
extent  by  the  rigidity  of  the  columns.  Were  the  columns  perfectly 
rigid,  the  proportionality  of  stress  to  deflection  would  be  unaffected 
by  the  columns,  but  in  case  they  participate  in  the  bending  of  the 
slab,  the  exactness  of  this  proportionality  will  not  be  entirely  pre- 
served. But  as  loads  increase,  a  divergence  will  appear  between 
the  rate  of  the  increase  in  stress  and  rate  of  the  increase  in  deflection. 

In  the  Norcross  test  slab,  where  the  column  effect  is  a  slight  un- 
resisted  tipping,  this  is  nearly  negligible,  amounting  under  load  4, 
to  nearly  three  percent,  while  under  Load  4,  Mushroom  test,  it- 
amounts  to  six  times  as  much,  or  to  an  even  eighteen  per  cent. 

The  large  variation  under  Load  4.  between  the  stress  ratio  and 
deflection  ratio  in  the  Mushroom  test,  indicates  an  intimate  relation 
of  column  stiffness  to  stress  and  deflection.  The  continuity  of  the 
slab  being  secured  by  the  integral  connection  of  the  column  and  slab, 
a  certain  amount  of  potential  energy  is  stored  within  the  columns  by 
flexure;  and  as  we  are  tracing  out  and  locating  all  the  leaks  in  our 
storage  system  of  energy,  these  columns  must  be  considered.  As 
restraint  of  connection  is  secured  in  part  by  the  columns,  any  column 
bending  reduces  the  amount  or  degree  of  fixity,  and  increases  the 


COMPARISON    OF    DEFLECTIONS  263 

slab  deflection  and  stress  at  mid  span.  But  increased  slab  deflection 
and  increased  stress  at  the  panel  center  involve  increase  of  work 
done  on  the  slab  and  columns  by  the  load,  in  greater  amount  than 
merely  in  proportion  to  the  increase  of  stress  in  the  slab  represented 
by  divergence  noted  in  the  increase  in  stress  and  in  deflection.  For 
consider  the  analogous  case  of  a  beam  fixed  horizontally  at  supports. 
If  these  supports  yield  sufficiently  to  cause  it  to  act  as  a  simple 
beam,  the  deflection  is  increased  five  times,  and  the  work  done  by 
the  load  is  multiplied  by  a  somewhat  larger  number  than  that. 
Similarly,  when  the  columns  supporting  a  slab  yield  somewhat  by 
tipping  they  must  increase  the  central  deflection  of  the  slab  and  so 
increase  the  energy  stored  in  the  slab,  in  addition  to  what  they  them- 
selves absorb  in  the  column  flexure  by  tipping.  The  energy  stored 
in  the  column  itself,  like  that  of  any  beam,  may  afford  a  leak  which 
is  a  fraction  of  its  energy  of  flexure,  and  tho  it  may  not  be  a  large 
fraction  of  the  total  energy  expended  upon  the  structure,  it  may 
nevertheless  be  the  cause  of  some  increase  in  the  total  energy  of 
deformation. 

The  jogs  in  the  seven  day  period  after  Load  4,  Fig.  70  and  in  the 
four  day  period  after  Load  5,  may  be  accounted  for  in  part  by  the 
fact  that  the  ground  was  frozen  and  there  was  perhaps  some 
heaving  affecting  the  bench  marks  at  which  the  levels  were  taken. 

In  the  curves  of  deflections,  Fig.,  70  the  influence  of  the  direct 
tensions  is  observed  in  the  slight  curvature  of  the  line  between 
Loads  1  and  3,  and  the  small  percent  of  loss  from  1his  source  is  in- 
dicated by  the  fact  that  the  curves  of  deflection  continue  parallel 
and  show  little  or  no  divergence  in  direction  under  higher  loads 
than  under  load  of  lower  intensity,  which  is  in  strong  contract  to 
curves  of  deflection  of  the  Norcross  test  slab. 

The  views  which  have  been  put  forth  in  the  foregoing  pages  to 
account  for  the  radical  experimental  difference  between  the  deport- 
ment of  beams  and  that  of  slabs  by  a  rigid  application  of  the  funda- 
mental laws  which  have  been  already  enunciated  have  been  vig- 
orously opposed  by  an  attempt  on  the  part  of  certain  members  of 
the  engineering  profession  to  explain  the  wide  divergence  of  actual 
slabs  from  the  results  of  beam  strip  theory  by  a  pretended  belief  in 
the  efficacy  and  sufficiency  of  the  direct  tensile  resistance  in  concrete 
as  sufficient  to  account  for  the  phenomena  observed  in  the  flexure 
of  slabs. 

In  order  to  determine  what,  if  any,  basis  in  fact  there  might  be 
for  any  such  view,  a  test  slab  twenty-five  feet  square,  and  approx- 


264  TENSILE    STRESSES    IN    CONCRETE 

imately  five  inches  thick,  was  constructed,  which  was  supported 
at  its  edges  by  walls  and  at  its  center  by  a  masonry  pier  20  inches 
square  and  reinforced  with  wire  netting  radiating  from  the  center 
in  the  bottom  of  the  slab.  The  netting  was  ordinary  poultry  netting 
and  2  inch  mesh,  galvanized.  The  applied  load  was  15,000  pounds 
in  the  form  of  concrete  barrels  partly  filled  with  water  and  arranged 
in  circular  formation  at  mid  span  around  the  central  pier.  When 
the  load  was  first  applied  the  slab  carried  the  load  by  tensile  resist- 
ance of  the  concrete  without  apparent  over  strain,  no  cracks  of  any 
kind  appearing.  The  load  was  left  in  place,  and  owing  to  the  leak- 
ing of  the  barrels  which  were  imperfect,  gradually  diminished  by  25 
or  30  percent.  In  the  course  of  about  five  days  cracks  began  to 
develop  in  the  slab,  these  extending  in  the  top  radially  and  circum- 
ferentially  about  the  center  pier  and  in  the  bottom  of  the  slab  at 
approximately  the  same  time  from  one  corner  along  mid  spans. 
The  slab  was  left  undisturbed  for  six  days  longer,  and  these  cracks 
continued  to  increase  until  finally  the  whole  structure  collapsed 
completely.  The  concrete  was  found,  after  the  collapse,  to  be 
nearly  5f"  thick  at  the  center  as  against  5  inches  at  the  edge. 

This  test  is  of  value  as  showing  somewhat  the  effect  of  time, 
combined  with  temperature  changes  upon  the  endurance  of  tensile 
stresses  in  the  concrete.  The  test  was  made  in  the  fall  of  the  year 
and  the  drop  in  temperature  from  mean  conditions  under  which 
the  slab  was  cured  may  be  stated  as  approximately  35  to  40  degrees. 

The  reinforcement  of  this  slab  was  designed  with  the  purpose  of 
making  it  substantially  like  the  wire  netting  of  certain  old  floors  of 
similar  span  which  had,  however,  a  thickness  of  from  12  to  15  inches 
the  outside  edges  of  these  latter  slabs  being  supported  vertically 
and  laterally  by  heavy  masonry  retaining  walls  which  formed  sub- 
stantial abutments  and  in  their  action  as  retaining  walls  caused  a 
certain  amount  of  thrust  to  act  upon  the  slab. 

The  great  thickness  of  these  floors  relative  to  span  caused  arch 
action  to  predominate  rather  than  slab  action  and  their  permanent 
stability  in  contrast  with  the  slab  tested  is  readily  accounted  for 
on  this  principle,  and  sharply  differentiates  slab  action  from  arch 
action. 

The  predominance  of  arch  action  is  dependent  upon  a  large  ratio 
of  thickness  to  span  and  vanishes  practically  in  a  thin  slab  of  long 
span. 

Mr.  Arthur  R.  Lord,  in  a  paper  published  in  the  " Engineering 
and  Contracting,"  January  29,  1913,  reports  interesting  data  rela- 


ARCH    ACTION    IN    SLABS  265 

tive  to  the  test  of  the  Larkin  Building  in  Chicago,  and  concludes 
that  there  is  a  marked  degree  of  arch  action  in  a  slab  the  span  of  which 
is  approximately  twenty  times  its  thickness.  He  reasons  that  there 
is  such  action  because  where  the  line  of  inflection  should  be,  he 
observed  compressions  in  the  concrete  both  in  the  top  and  bottom  of 
the  slab,  and  infers  from  this  that  these  compressions  are  a  measure 
of  arch  action.  Now,  it  is  a  fundamental  principle  of  flexure  that 
the  sum  of  the  horizontal  compressions  must  equal  the  sum  of  the 
tensions  at  any  section  thru  a  plate  in  bending,  if  no  arch  action  be 
present.  Examining  the  cross  section  of  this  design,  we  find  the  slab 
rods,  at  the  deflection  line,  crossing  the  neutral  plane  of  the  slab 
at  an  angle,  and  turning  downward  at  a  considerable  inclination. 
Now  since  there  is  shear  across  this  section,  these  rods  must  be  in 
tension  at  the  line  of  inflection,  by  virtue  of  vertical  shear,  and  the 
horizontal  component  of  the  tension  in  the  steel  must  be  balanced, 
to  fulfill  the  laws  of  flexure,  by  compressions  in  the  concrete;  and  hence 
this  supposed  arch  action  is  thus  readily  accounted  for  as  a  phe- 
nomena of  flexure.  Moreover  the  magnitude  of  the  thrust  was 
wholly  insufficient  to  account  for  the  carrying  capacity  of  the  slab 
in  excess  of  beam  theory. 

4.  Investigation  of  Structures  by  the  Berry  Extensometer  and 
Interpretation  of  Results.  In  the  investigation  of  concrete  struc- 
tures with  the  Berry  Extensometer,  or  similar  instrument,  it  is  usually 
possible  to  secure  measurements  on  one  side  only  of  the  reinforcing 
rod,  and  hence  the  measurement  is  primarily  a  measurement  of 
fiber  stress  rather  than  that  of  the  average  stress  across  the  section 
of  the  bar.  Any  kink  in  the  bar,  due  to  careless  handling  before  plac- 
ing in  the  structure,  is  liable  to  induce  a  bending  stress  under  work- 
ing conditions,  which  will  mask  in  a  large  measure  the  character 
of  the  average  stress  in  the  bar  and  its  true  mechanical  action  in  the 
structure.  This  difficulty  might  be  obviated  if  we  could  get  at  both 
the  top  and  bottom  of  the  bar,  and  take  observations  on  both  the 
lower  and  upper  fibers;  but  it  is  generally  impracticable  to  do  this, 
and  far  better  to  check  up  the  accuracy  of  the  readings  by  careful 
comparison  with  observed  deflections. 

The  next  difficulty  in  the  experimental  solution  of  the  problem 
of  stresses  in  flat  slabs  by  the  strain  gage  lies  in  the  masking  of  the 
true  action  of  the  material  by  the  stresses  induced  in  the  process  of 
casting.  These  stresses  naturally  vary  thru  a  wide  range,  dependent 


266  CONDITIONS    AFFECTING    EXTENSOMETER    TESTS 

on  the  temperature  conditions  at  which  the  concrete  was  cast  and  the 
temperature  conditions,  humidity  and  barometic  pressure  of  the  air 
under  which  the  concrete  was  cured. 

In  practical  work,  it  is  frequently  the  case  that  where  the  work 
is  executed  in  hot  weather,  the  steel  and  the  concrete  materials 
are  heated  by  the  sun  and  are  quite  warm  when  the  con- 
crete is  mixed,  so  that  very  rapid  setting  and  hardening  results. 
Such  hardening  is  accompanied  in  the  chemical  process  of  curing 
with  a  considerable  evolution  of  heat,  and  the  steel  is  thus  heated 
to  a  temperature  as  high  as  130°  Fahrenheit  or  even  more  during  this 
process.  If  the  hardening  is  sufficiently  rapid  to  form  a  rigid  bond 
between  the  steel  and  the  concrete  during  this  stage  of  hardening, 
(and  it  frequently  does  form  such  bond)  the  final  result  is  that  as  the 
mass  cools  down,  the  steel  is  thrown  into  tension  by  the  cooling  and 
the  concrete  to  a  considerable  degree  into  compression,  this  com- 
pression being  distributed  over  the  cross  section  of  the  slab.  The 
result  of  the  combined  temperature  and  shrinkage  stresses  induced 
in  hot  weather,  is  such  as  at  times  to  cause  the  slab  to  be  practically 
self  supporting  and  remove  its  weight  largely  or  entirely  from  the 
supporting  forms  so  that  these  in  hot  weather  are  frequently  found 
to  be  really  loose,  and  may  be  knocked  out  with  little  resistance. 

This  condition,  to  some  extent,  may,  of  course,  be  accounted  for 
by  the  shrinkage  of  the  lumber  forms  which  are  wet  in  casting,  but 
this  shrinkage  is  insufficient  to  account  for  the  difference  in  condi- 
tions observed  in  warm  weather  work  contrasted  with  cold  weather 
work. 

The  presence  of  such  shrinkage  stresses  in  the  material,  cause 
its  apparent  deportment  to  be  materially  different  under  loads  of 
low  intensity  from  its  action  under  loads  of  higher  intensity  where 
the  mechanical  operation  of  the  combination  is  not  masked  by 
extraneous  influences. 

The  effect  of  casting  stresses  and  shrinkage  stresses  which  have 
been  referred  to  above  gradually  disappears  of  course  with  time  and 
continuance  of  the  chemical  process  of  hardening,  and  under  the 
repeated  changes  in  form  of  the  structure  caused  by  temperature 
variations  and  changes  of  load.  Accordingly  it  must  be  kept  clearly 
in  mind  that  positive  conclusions  as  to  the  mechanical  operation 
of  the  slab  cannot  be  deduced  under  loads  that  are  too  small  to  per- 
mit the  character  of  the  stresses  induced  by  the  load  to  be  distin- 
guished from  stresses  originally  induced  by  the  weather  conditions 
while  casting. 


CONDITIONS    AFFECTING    EXTENSOMETER    TESTS  267 

Measured  stresses  on  newly  cured  work  can  be  given  weight, 
accordingly,  only  after  the  loads  applied  become  materially  greater 
than  the  working  load.  The  true  action  of  the  structure  commences, 
then,  to  become  dominant,  because  this  action  is  not  masked  by  the 
influence  above  discussed.  These  influences  have  been,  by  some, 
improperly  credited  to  an  impossible  direct  tensile  resistance  of 
concrete. 

Moreover,  measurements  of  the  deformations  in  the  concrete 
by  the  extensometer  are  frequently  erroneously  interpreted.  In 
the  practical  testing  of  a  building  applied  loads  remain  upon  the 
concrete  a  considerable  period  of  time,  since  a  comprehensive  survey 
of  the  stresses  in  the  slab  cannot  be  executed  short  of  several  day's 
continuous  work.  When  it  is  attempted  to  interpret  extensometer 
measurements  of  the  concrete  which  has  been  subjected  to  a  given 
load  continuously  for  several  days  or  a  week  on  the  basis  or  the 
modulus  of  elasticity  determined  by  measurements  made  on  test 
cylinders  of  concrete  which  are  loaded  with  given  loads  for  very  short 
periods  only  a  considerable  error  is  involved  in  such  comparison. 
First,  because  an  8  inch  cylinder  16  inches  long  cast  at  the  same  time 
that  the  floor  of  the  building  was  cast,  has  a  better  opportunity  to 
dry  out  and  become  hard  and  rigid  before  testing  than  the  concrete 
work  of  the  practical  structure.  Second,  because  the  short  period  of 
time  in  which  the  load  is  applied  to  the  cylinder  in  the  ordinary 
method  of  making  tests  does  not  correspond  to  the  time  element 
involved  in  making  tests  of  the  work  in  the  finished  structure,  and 
neglect  of  these  conditions  involves  a  fundamental  error  lost  sight 
of  frequently  in  the  experimental  determination  of  concrete  stresses. 
The  correct  method  would  be  to  determine  the  residual  set  of  the 
concrete  prism  under  a  continued  load  of  the  intensity  which  it  is 
desired  to  interpret,  then  deduct  this  set  from  the  measurements 
made  on  the  practical  structure  and  determine  the  true  modulus 
of  the  specimen  by  repeated  loadings.  Scientific  results  may  be 
thus  secured  which  would  be  of  value  in  checking  the  mathematico- 
elastic  theory. 

The  great  difficulty  with  extensometer  tests  aside  from  the 
labor  and  expense  involved,  is  due  to  the  great  uncertainty  arising 
from  the  causes  which  have  been  mentioned  and  to  the  fact  that 
measurements  taken  on  corresponding  rods  at  corresponding  points 
where  like  results  would  be  expected  differ  so  greatly  as  to  show 
that  accidental  differences  of  construction  have  so  large  an  effect 
upon  the  measurements  as  to  make  precise  deductions  very  difficult. 


268  UNCERTAINTITIES    OF    EXTENSOMETER    TESTS 

Such  inequalities  would,  however,  evidently  not  be  dangerous  to  the 
structure  because  overstrain  on  any  rod  would  ultimately  be  relieved 
by  others  coining  into  action  nearby. 

The  measurement  and  interpretation  of  deflections  under  load 
is  not  beset  by  uncertaintities  and  difficulties  of  this  character.  A 
deflection  is  the  result  of  the  combined  action  of  all  the  elements  of 
the  slab  and  not  of  any  single  one  exclusively  and  so  has  a  degree 
of  reliability  which  cannot  attach  to  any  result  derived  from  measure- 
ments on  single  elements  however  numerous.  If  deflections  and 
stresses  are  mathematical  elements  of  a  comprehensive  slab  theory 
the  measurement  of  either  one  is  sufficient  to  determine  the  other 
just  as  in  the  theory  of  beams.  When  the  profession  shall  have  be- 
come convinced  of  the  validity  and  sufficiency  of  slab  theory,  there 
will  be  little  use  for  extensometer  tests.  Deflections  are  sufficient. 


269 


CHAPTER  VII 
MOMENTS  IN  TWO-WAY  AND  FOUR-WAY  FLAT  SLABS 

1.  Simple  Approximate  Theory  of  Four- Way  Slabs.  In  order- 
to  investigate  approximately  the  applied  bending  moments  and 
resulting  stresses  in  a  four-way  flat  slab  in  a  more  elementary  man- 
ner and  dispense  with  the  use  of  higher  mathematics,  assume  that 
each  of  the  four-way  reinforcing  belts  supports  one-fourth  of  the 
total  uniformly  distributed  panels  loads  W.  This  is  very  nearly 
the  fact  in  the  central  portion  of  the  slab  where  the  curvature  is 
concave  upward  and  the  side  and  diagonal  belts  are  to  a  considerable 
extent  separate  from  each  other. 

Assume  that  the  central  portion  of  each  side  belt  for  example 
at  least  as  far  as  the  lines  of  inflection,  is  uniformly  loaded  with  a 
part  of  W  /4  proportional  to  its  length  and  that  the  position  of  the 
lines  of  inflection  is  the  same  as  would  be  found  in  a  uniform  can- 
tilever beam,  viz:  at  a  distance  \L  /'\/3  =  .288L  each  way  from  mid 
span.  The  assumption  however  that  so  far  as  the  central  portion 
of  the  side  belt  is  concerned  the  load  W  /4  may  be  taken  as  uniformly 
distributed  is  only  approximate,  for  the  load  is  concentrated  some- 
what toward  mid  span  as  may  be  seen  from  Fig.  73  where  the  load 


\ 

If 


A' 


Fig.  73. 


upon  half  a  side  belt  of  a  panel  may  be  taken  as  that  resting  on  the 
triangle  A  B  B",  and  that  upon  half  a  diagonal  belt  as  that  on  the 
quadrilateral  A  B  C  B' .  This  assumes  that  there  are  no  vertical 
shearing  stresses  in  the  slab  on  the  lines  A  B,  A  B'  etc.,  which  would 
not  necessarily  be  exactly  the  fact,  especially  for  oblong  panels. 


270  POSITION    OF    LINES    OF    INFLECTION 

But  the  lines  of  inflection  have  been  assumed  above  to  be  at  some- 
what greater  distances  from  mid  span  than  occurs  in  a  slab  where 
the  caps  have  a  diameter  of  0.2L  or  more,  so  that  the  applied  moment 
at  mid  span  of  a  uniformly  loaded  continuous  beam  of  length  L  will  be 
approximately  that  of  a  side  belt  in  a  standard  mushroom  slab.  It 
should  however  be  noticed  that  the  position  of  the  lines  of  inflection 
is  a  matter  which  is  determined  by  the  designer  and  within  practical 
limits  is  within  his  control,  tho  so  far  as  known  they  are  universally 
assumed  to  occur  where  they  would  be  situated  in  a  plate  which 
opposes  to  the  applied  moment  a  uniformly  distributed  moment 
of  inertia.  This  is  not  the  case  with  a  reinforced  cantilever  slab, 
any  more  than  it  is  with  a  continuous  cantilever  bridge  where  the 
moment  of  inertia  is  reduced  to  zero  at  the  ends  of  the  suspended 
span  by  joints.  The  resisting  moment  of  inertia  is  practically 
reduced  to  zero  in  the  reinforced  slab  at  the  lines  where  the  rein- 
forcement dips  below  the  neutral  axis  and  thus  the  lines  of 
inflection  are  fixed  at  these  loci. 

Designs  which  have  definite  bends  in  the  slab  rods  where  they 
make  a  somewhat  steep  descent  from  the  top  to  the  bottom  of  slab 
are  to  be  avoided,  for  any  severe  stress  at  such  a  bend  is  apt  to  make 
cracks  in  the  concrete,  while  there  is  nothing  in  slab  construction 
to  forbid  a  very  gradual  dip  from  top  to  bottom  of  slab  at  the  lines 
of  inflection  where  the  moments  gradually  approach  zero. 

But  designs  in  which  some  of  the  belt  rods  dip  suddenly  at  one 
distance  from  the  column  center,  and  others  at  a  different  distance 
and  still  others  at  another  distance,  are  especially  reprehensible 
because  they  mechanically  obliterate  any  definite  lines  of  inflection 
and  put  them  in  a  different  position  for  each  different  unbalanced 
load,  and  so  introduce  uncertainty  in  place  of  certainty  in  design. 
Especially  is  this  true  in  case  of  any  accidental  subsidence  of  column 
under  load  where  the  same  principles  obtain  as  in  a  cantilever  bridge 
with  joints,  as  compared  with  a  continuous  bridge,  where  the  latter 
is  liable  to  dangerous  stresses  in  case  of  subsidence  from  which  the 
cantilever  is  measurably  free. 

The  designer  of  a  slab  thus  having  control  of  the  size  of  his  can- 
tilever and  consequently  of  the  position  of  his  lines  of  inflection 
naturally  removes  these  lines  as  far  as  circumstances  will  permit 
from  column  centers  when  by  so  doing  the  stresses  in  the  concrete 
around  the  column  cap  are  not  too  greatly  increased. 


VALUE    OF    K   IN    SLABS  271 

Assume  in  the  first  place  for  the  purposes  of  computation  that 
the  part  of  the  side  belt  lying  brtween  the  points  of  inflection  is  a 
simple  beam  of  length  0.577L,  loaded  with  a  proportionate  load 
of  0.577TF  /4.  Then  the  applied  moment  at  its  center  will  be  (as 
in  any  simple  beam)  one  eighth  the  product  of  these  quantities,  viz : 
(0.577)2  W  L  /32  =  W  L  /96. 

In  order  to  derive  from  this  applied  moment  the  resisting  moment 
of  the  rods  of  the  side  belt  the  effect  of  the  diagonal  belts  which  cross 
the  side  belts  diagonally  and  increase  the  cross  section  of  the  steel 
resisting  the  moment  by  about  50  per  cent  on  the  average  must  be 
allowed  for,  as  well  as  the  mutual  effect  of  the  reinforcing  rods 
which  cross  each  other  under  tension  at  the  edges  of  the  belt  and 
the  embedment  which  have  an  effect  to  reduce  the  stresses  in  the 
side  belt,  an  effect  which  is  dependent  upon  Poisson's  ratio  K. 
thru  the  action  of  bond  shear  already  discussed  in  this  paper.  As 
already  shown  in  Chapter  V,  both  the  stresses  in  the  steel  and  the 
deflection  in  the  slab  are  reduced  in  general  by  the  factor  (1 — K2)  in 
case  of  a  Poisson  ratio  =  K. 

No  direct  determinations  of  K  for  such  a  composite  material  as 
reinforced  concrete  are  available,  but  every  test  for  stresses  or  deflec- 
tion may  be  regarded  in  the  light  of  a  determination  of  K  provided 
the  formulas  for  these  quantities  are  completely  known  otherwise. 

A  general  value  of  K  =  0.5  brings  a  good  agreement  between 
the  formulas  previously  given  and  the  observed  data  in  a  very 
large  number  of  tests,  some  of  which  have  already  been  detailed 
in  Chapter  VI  entitled  Steel  Stresses  in  Flat  Slabs.  It  is  probable 
that  K  would  have  somewhat  different  values  for  different  arrange- 
ments of  belts  with  reference  to  each  other.  Now  the  value  of 
K  =  0.5  is  one  which  would  necessarily  hold  for  any  incompressible 
solid  i.  e.,  a  solid  of  constant  volume,  while  K=l  is  a  value  which 
would  apply  to  a  sheet  of  constant  area  without  regard  to  thickness. 
Great  objection  has  been  raised  to  adopting  so  large  a  value  of  K 
as  0.5  but  its  total  effect,  depending  as  it  does  upon  the  factor 
(l — K2),  is  at  most  to  make  the  stress  75  percent  of  what  it  other- 
wise would  be  and  that  is  believed  not  to  be  an  over  estimate  of 
the  effect  of  the  bond  shear  which  has  been  previously  discussed. 

Introducing  therefore  the  effect  of  the  increase  of  the  amount  of 
reinforcement  due  to  the  overlapping  of  the  diagonal  belts,  and  also  that 
due  to  the  lateral  effect  into  the  expression  for  the  part  of  the  resisting 
moment  exerted  by  the  direct  rods  in  the  side  belts  it  becomes 

2  /3  (1— K2)  W  L  /96  -  W  L  /192 .  .  .  .  (1) 


272  IRREGULARITIES    IN    DESTRUCTION    OF    STEEL 

in  which  the  first  factor  takes  account  of  the  fact  that  the  direct 
rods  constitute  on  the  average  only  two  thirds  of  the  tension  rein- 
forcement actually  present,  and  (1 — X2)=0.75  if  /\=0.5,  takes 
account  of  the  reduction  of  stress  due  to  the  lateral  action  expressed 
by  Poisson's  ratio.  This  is  precisely  the  same  result  that  was 
reached  in  equation  (34a)  Chapter  V,  which  was  derived  by  the 
application  of  exhaustive  mathematical  analysis  to  a  continuous 
uniform  slab  square  or  oblong  and  supported  at  the  corners,  where 
L  is  the  length  of  the  side  belt  under  consideration. 

There  is  one  other  question  in  this  connection  which  needs  con- 
sideration, viz:  the  irregularity  of  the  lapping  of  the  belts  over  the 
area  of  the  side  belts.  The  question  is  as  to  what  amount  of  irregu- 
larity of  distribution  may  exist  without  materially  interferring  with 
or  changing  the  action  of  the  total  amount  of  steel.  All  designers 
and  investigators  agree  that  a  belt  of  rods  is  practically  equivalent 
in  its  action  lengthwise  of  the  rods  to  a  sheet  of  metal  of  equal 
width  and  weight  if  all  questions  of  bond  be  disregarded,  and  the 
question  is  whether  other  large  irregularities  of  distribution  such  as 
occur  in  the  over-lapping  of  side  and  diagonal  belts  may  be  disre- 
garded, and  whether  the  mean  weight  of  metal  present  is  the  only 
significant  factor.  Such  would  seem  to  be  the  fact  within  limits 
of  area  which  are  comparatively  small  fractions  of  the  total  panel 
area.  This  may  be  stated  more  convincingly  perhaps  by  saying  that 
it  is  impossible  to  elongate  the  central  portions  of  the  side  belts  with- 
out at  the  same  time  elongating  the  steel  of  the  diagonal  belts  that 
lies  along  the  edges  of  the  side  belt.  Tests  show  what  is  otherwise 
evident  that  the  elongations  in  all  the  rods  across  the  side  belts  are 
practically  the  same.  Hence  the  diagonal  rods  at  the  edges  of  the 
side  belts  participate  in  the  same  elongations.  And  this  is  the 
basis  of  the  assumption  of  an  average  reinforcement  of  50  percent 
in  addition  to  the  side  belts  themselves. 

Next  compute  the  moments  at  the  middle  of  the  diagonal  belts, 
each  under  a  total  assumed  load  of  W  /4  uniformly  distributed.  If 
the  distance  between  inflection  points  on  the  diagonal  be  taken  to 
be  2  times  that  on  the  side  belts,  then  the  applied  moment  at  the 
center  becomes  W  L  \2  /96  and  the  resisting  moment  of  the  steel 
at  mid  span  in  one  diagonal  belt  may  be  written 

i\2~(l—  K2)  W  L/96  =  IFL/180 (2) 

in  which  the  factor  |  takes  account  of  the  fact  that  one  diagonal 
belt  comprises  only  one  half  of  the  reinforcing  steel  present,  and 
(1 — K2)  takes  account  of  the  reduction  of  stress  due  to  the  lateral 


MOMKNTS    IN    KOUK-WAY    SLAM  27'.") 

action  expressed  by  Poisson's  ratio.  No  account  however  has  been 
taken  of  the  reduced  concentration  of  the  load  at  mid  span  of  the 
diagonals  as  shown  in  Fig.  73,  which  in  fact  makes  the;  stresses  at 
mid  span  of  the  diagonal  belts  not  only  less  than  those;  computed 
from  (2)  but  somewhat  less  even  than  those  computed  from  (1)  for 
the  side  belts,  a  fact  which  is  established  by  the  observed  results 
of  all  available  tests  of  four-way  slabs  in  buildings.  The  same 
fact  appears  mathematically  from  the  results  of  the  more  exact 
analysis  given  in  Chapter  V,  so  that  with  equal  belts  in  four-way 
reinforcement  greater  stresses  occur  at  mid  span  of  the  sick;  belts 
than  at  the  center  of  the  panel  in  the;  diagonals. 

Consider  in  the  next  place;  the  applied  moments  at  the  column 
heads.  In  a  uniformly  loaded  cantilever  beam  such  as  has  been 
assumed  for  the  purposes  of  computation,  each  side  belt  will  have 
an  applied  moment  at  each  end  which  is  twice;  that  at  mid  span 
viz.:  W  L  /48,  making  a  total  applied  moment  for  the;  four  belts  in 
180°  around  the  column  center  of  W  L  /12.  Owing  to  the  some- 
what greater  concentration  of  stresses  in  the;  center  rods  of  the  belt 
by  reason  of  their  being  at  a  level  above;  those  at  the  eielges  of  the1 
beilt,  as  well  as  by  reason  of  the'  concentration  erf  stress  at  the;  mielelle' 
rods  of  the  belt  at  the  enlge1  of  the  cap  elue;  to  its  rigielity  the;  ele>cre>ase 
of  belt  stresses  arising  from  the  shortening  of  the  cle>ar  span  by  the; 
c-aps  will  be>  disregarded  in  obtaining  this  roughly  appre>ximate' 
value  of  the  stresses  at  the  edge  of  the  cap.  Disre;gareling  then-fore 
any  reduction  of  the  moment  elue;  to  shortening  the;  span  by  the; 
breadth  of  the  support  afforded  by  the  column  caps  anel  assuming 
that  each  belt  is  carried  across  the  column  as  a  continuous  be'arn 
the'  question  arises  as  to  what  reduction  of  stress  will  arise;  from 
other  steel  with  which  it  is  in  contact  by  its  coaction  therewith. 
Assume  as  a  safe  basis  of  computatiem  that  e>ach  be'lt  coacts  with 
one  other  belt  as  elo  each  of  the'  diagonal  belts  at  the  pane-1  center. 

The'  resisting  moment  of  each  side'  be'lt  at  the;  edge'  of  the'  cap 
will  then  be  written 

i  (1—  K2)  WL/4$  =  W  L/128 (3) 

in  which  the  factor  \  takes  account  of  the;  ste;e;l  other  than  the  belt 
itself  in  assisting  the  belt,  anel  (1 — K2)  give's  the  additional  reduction 
due  to  the'  lateral  action  in  the  slab  of  Poisson's  ratio. 

From  this  it  is  eviele'nt  that  with  steel  at  the  same  distance  fre>m 
the  neutral  axis,  50  per  ce'nt  more  ste'e'l  woulel  be  required  according 
to  this  computation  in  each  belt  over  the;  column  he'ael  than  at  mid 
span,  which  increase  is  to  be  provieleel  for  by  laps  or  otherwise  in 


274  TWO-WAY    REINFORCEMENT 

the  belts  over  the  head.  But  these  laps  need  not  be  distributed  equally 
among  the  belts.  Any  or  all  the  laps  may  occur  equally  well  in  two 
belts  only  by  extra  rods  placed  between  the  belt  rods  which  indi- 
vidually extend  several  spans.  The  laps  or  extra  rods  will  be  more 
effective  the  nearer  they  are  to  the  top  of  the  slab  and  also  the 
nearer  they  are  to  the  middle  of  the  belt,  because  the  edges  of  the 
belts  are  at  a  somewhat  lower  level  than  the  middle  of  the  belts. 

The  steel  of  the  ring  and  radial  rods  has  been  left  out  of  the 
account  in  this  rough  computation  as  well  as  the  breadth  of  the 
cap,  in  order  to  offset  the  smaller  arm  with  which  part  of  the  belts 
act  when  cob  piled  one  on  another  at  the  top  of  the  column  as  well 
as  to  compensate  for  the  lower  level  of  the  belts  at  their  edges. 

2.  Simple  Approximate  Theory  of  Two-Way  Slabs.  To  investi- 
gate in  a  similar  manner  the  flat  slab  with  two-way  reinforcment 
suppose  the  lines  of  inflection  to  be  situated  as  before  at  a  distance 
of  .288L  each  way  from  mid  span.  Then  the  width  of  the  central 
area  between  lines  of  inflection  is  .577L  and  the  width  of  the  side 
belts  is  .423L. 

Let  the  loading  upon  each  central  area  of  a  panel  between  the 
side  belts  be  transmitted  symmetrically  sidewise  to  the  side  belts 
by  the  median  belts.  Each  central  median  belt  parallel  to  the  sides 
may  be  regarded  as  constituting  a  simple  beam  of  length  .577  L  and 
carrying  a  uniform  load  of  W  /6  or  half  that  on  this  central  area 

There  will  consequently  be  a  positive  central  applied  moment 
in  each  median  belt  at  mid  span  amounting  to  one  eighth  the  product 
of  the  load  and  span,  which  is 

1  W  L      WL 


The  resisting  moment  of  the  steel  in  one   median  belt  at  mid 
span  will  be 

J  (1—  K2)  WL/83  =  WL/222  .....................  (4) 

Where  the  factor  J  takes  account  of  the  fact  that  one  belt  is  only 
half  of  the  reinforcement  present  and  (1  —  -K2)  makes  allowance  for 
lateral  action  of  the  other  belt.  This  will  give  the  mean  stress  in 
the  slab  rods  of  the  median  belt.  The  middle  rods  of  this  belt 
however  have  greater  stresses  than  this.  The  negative  moment 
applied  to  the  median  belt  across  the  edge  of  the  panel  at  the  middle 
of  the  side  belt  will  be  one  eighth  of  the  product  of  its  load  W  /6 
by  the  width  of  the  side  belt  regarded  as  the  length  of  the  simple 
beam  transmitting  this  load  to  the  side  belt  and  uniformly  sup- 


MEDIAN    AND    SIDE    BELTS  275 

ported  by  it.     Hence  the  moment  is 

1       W   A2'3L=WL 
8       6  112 

This  is  also  the  resisting  moment  of  the  median  belt  at  this  point 
because  there  is  no  steel  in  the  top  of  the  slab  coacting  with  it. 
This  resisting  moment  is  the  greatest  in  this  belt.  It  consequently 
determines  the  cross  section  of  the  steel  in  the  entire  belt  which 
should  not  be  less  at  the  panel  center  since  the  stress  in  the  middle 
rods  at  the  panel  center  is  greater  than  at  the  edges.  The  effect 
of  the  median  belts  is  to  transfer  that  portion  of  the  load  actually 
covering  the  central  area  between  the  side  belts,  viz.  W  /6,  and  place 
it  upon  those  belts,  so  that  the  load  acting  upon  each  side  belt  of 
length  L  between  columns  is  %W,  irrespective  of  its  width  and  the 
size  of  the  central  area. 

It  will  be  assumed  that  this  load  is  uniformly  distributed  along 
the  side  belt,  tho  its  apparent  distribution  has  a  somewhat  greater 
concentration  toward  mid  span,  as  may  be  seen  by  considering  the 
situation  of  the  square  areas  included  between  the  panel  diagonals 
of  several  panels,  for  on  drawing  these  diagonals  the  square  load 
areas  supported  by  each  side  belt  have  corners  at  column  centers 
and  at  panels  centers.  The  median  belts  will  have  some  effect  to 
transmit  loads  diagonally  as  well  as  laterally  and  it  is  not  far  from 
correct  to  assume  uniform  distribution  of  load  upon  the  side  belts, 
tho  that  assumption  reduces  their  central  moments  somewhat,  as  was 
the  case  to  a  less  extent  for  the  side  belts  of  the  four- way  slab.  With 
twice  the  load  of  the  side  belt  of  the  four-way  slab  upon  each  side 
belt  of  the  two-way  slab  the  applied  moment  at  mid  span  of  each 
side  belt  will  be  twice  that  in  the  four-way  side  belt,  viz.:  W  L  /48 
at  mid  span  of  a  side  belt.  This  is  also  equal  to  the  moments  of  re- 
sistance of  the  steel  in  the  side  belt  without  the  benefit  of  any  assis- 
tance from  the  steel  that  crosses  this  belt.  There  is  no  such  assis- 
tance here  because  the  median  steel  that  is  in  tension  lies  across  the 
top  surface  of  slab  and  cannot  coact  to  any  appreciable  extent  with 
the  steel  of  the  side  belt  at  the  bottom,  neither  can  it  coact  with 
any  steel  that  might  be  continued  across  the  bottom  in  compression 
as  is  sometimes  done.  The  applied  bending  moment  in  each  side 
belt  where  it  crosses  the  column  center  may  be  assumed  to  be  twice 
that  at  mid  span,  viz.:  W  L  /24  giving  a  total  moment  of  W  L  /12 
in  180  about  the  column  center. 

The  resistance  afforded  by  the  steel  in  each  belt  at  the  support  com- 
bined with  the  lateral  action  of  that  crossing  it  at  right  angles  will  be 


276  TWO-WAY    AND    FOUR-WAY    STEEL    COMPARED 

i  (l—K2)  W  L  /<2±  =  W  L  /64 (5) 

which  is  the  same  as  that  in  two  belts  of  the  four-way  slab.  This 
requires  less  steel  in  the  belt  where  it  crosses  the  top  of  the  column 
than  at  mid  span  and  permits  a  fraction  of  the  side  belts  rods  to  be 
carried  thru  on  the  bottom  of  the  slab  at  the  column  when  so  desired 
It  would  not  be  good  practice  to  reduce  the  total  cross  section  of  the 
side  belts  at  the  columns,  below  that  required  at  mid  span,  whatever 
theory  may  be  accepted  respecting  shearing  stresses  in  reinforcing 
rods  around  the  columns. 

3.  Weight  of  Steel  in  Two-Way  and  Four-Way  Slabs  Com= 
pared.  In  making  a  comparison  of  the  weight  of  slab  steel  required 
in  a  two-way  panel  with  that  in  a  four-way  panel  of  the  same  size 
and  thickness,  it  will  be  noticed  that  the  cross  section  of  the  steel 
required  in  each  belt  will  be  proportionl  to  its  resisting  moment  and 
its  weight  will  be  proportional  to  the  product  of  cross  section  by 
length. 

Now  omitting  common  factors  of  W  and  L  the  weights  will  be 
proportional  to  the  following  numbers: 

In  a  four-way  panel : 

Two  side  belts  together  give  by  (3) 2  /128 

Two  lapped  belts  half  length  give 1  /128 

Two  diagonal  belts  give 2\'2  /128 

Making  a  total  of  about 1  /22 

In  a  two-way  panel : — 

Two  side  belts  together  give 2  /48 

Two  median  belts  together  give 2/112 

Making  a  total  of 1  /16.8 

exclusive  of  laps.  This  shows  an  excess  of  weight  of  belt  rods  in 
the  two-way  panel  of  somewhat  more  than  30  percent  over  that 
required  in  the  four-way  panel,  but  does  not  take  account  of  any 
head  steel  used  in  supporting  the  belts  in  the  two-way  panel  over 
the  heads  of  the  columns,  nor  of  the  Mushroom  heads  in  the  four- 
way  panel. 

The  above  simplified  analysis  shows  how  this  excess  arises  in  the 
main,  viz.:  from  lack  of  suitable  arrangements  in  the  two-way 
reinforcement  to  take  advantage  of  coaction  of  belts,  and  besides 
that  the  excess  due  to  the  round  about  indirect  transmission  of  the 
loads  thru  the  median  belts  to  side  belts  instead  of  direct  transmission 
to  columns  thru  diagonals. 


TWO    AND    FOUR-WAY    SLABS    ON    WALL    SUPPORTS  277 

The  question  -of  the  weight  of  steel  required  in  a  20'  by  20'  panel 
designed  to  carry  a  total  live  and  dead  load  of  300  Ibs.  per  square 
foot  has  been  discussed  recently  by  the  writer*,  and  it  is  shown  that 
such  a  panel  would  require  about  1000  Ibs.  of  steel  according  to  sev- 
eral authorities  on  slab  design,  while  several  others  who  would 
reject  the  foregoing  slab  theory  as  inadvisable  and  insist  on  beam 
theory  as  alone  applicable  to  slabs  and  essential  for  safe  design 
would  require  about  2,000  Ibs.  of  steel  per  panel. 

It  has  been  tacitly  assumed  in  the  foregoing  computations  and 
comparisons  that  reinforcing  rods  across  the  top  of  the  side  belts 
in  four-way  slabs  are  unnecessary  and  superfluous,  and  that  the  cracks 
occasionally  observed  extending  along  the  middle  of  the  side  belts 
do  not  indicate  any  structural  weakness.  Such  is  the  fact,  since  the 
necessary  reinforcing  steel  to  reisst  the  negative  moment  occurring 
across  the  side  belts  is  to  be  found  in  the  parallel  side  belts  across 
the  column  heads.  The  cracks  where  they  exist  allow  sufficient 
deformation  and  twisting  moment  to  act  in  the  slab  to  bring  this 
belt  steel  into  play  in  this  way.  Cross  reinforcement  on  top  of  the 
side  belts  viewed  from  the  standpoint  of  mechanics,  only  serves  to 
increase  the  load  upon  them  and  so  increase  the  stress  in  them,  and 
at  the  same  time  relieves  to  some  extent  the  stress  in  the  diagonals, 
thus  making  the  method  of  slab  operation  to  resemble  the  uneco- 
nomical action  of  two  way  reinforcement. 

4.     Panels   Reinforced   Unequally   Lengthwise   and    Crosswise. 

The  particular  solution  of  the  general  partial  differential  equation 
(20)  of  Chapter  V  which  was  developed  in  that  Chapter  was  one 
that  has  special  reference  to  slabs  resting  on  separate  sup- 
ports or  columns  at  the  corners  of  the  panels.  It  is  a  solution 
in  which  the  deflections  at  mid  span  of  the  sides  of  a  square  panel 
are  more  than  half  as  great  as  at  the  panel  center,  and  one  in  which 
the  ratio  of  the  deflections  at  mid  span  of  the  sides  of  a  rectangular 
panel  varies  as  the  fourth  power  of  the  lengths  of  the  sides  so  that 
for  the  extreme  case  of  L2  /L\  =  .75  the  deflection  at  mid  span  of 
the  long  side  would  be  more  than  three  times  that  on  the  short 
side.  It  is  evident  therefore  that  such  a  solution  as  that  is  entirely 
inapplicable  to  the  case  of  a  slab  where  the  edges  of  the  panels  are 
supported  on  walls  which  deflect  not  at  all  or  on  beams  which  are  so 
stiff  that  their  deflections  are  small  compared  with  slab  deflec- 
tions. 


*  Proc.  Am.  Soc.  C.  E.,  May,  1914,  p.  1513. 


278  GRASHOFS'    PROPOSED    EQUATION 

Grashof  has  proposed  the  following  equation  as  the  best  he  was 
able  to  invent  to  represent  the  surface  of  the  middle  layer  of  a  uni- 
form rectangular  plate  fixed  horizontally  at  the  edges : 

24#/(a4+64)  z  =  q  (a2~x2)2 (b2—y2)2 (7) 

This  equation  was  proposed  by  Grashof  on  the  analogy  of  the  equa- 
tion for  a  beam  with  fixed  ends,  and  not  as  a  solution  of  the  differ- 
ential equation,  which  in  fact  it  is  not,  tho  it  has  often  been  quoted  as 
if  it  were  in  some  way  so  affiliated  with  the  differential  equation  as 
to  derive  some  validity  from  it.  Such  is  not  the  fact  however. 
No  solution  of  this  differential  equation  of  the  fourth  order  can 
contain  terms  of  higher  degree  than  the  fourth,  since  otherwise 
the  last  member  would  not  be  constant.  We  can  dismiss  Grashof's 
equation  as  simply  an  invention  of  an  ideal  nature.  He  was  aware 
that  the  equation  sought  must  contain  the  two  quantities  (a2—x2)2 
and  (b2—y2)2,  and  he  in  fact  proposed  that  the  result  contain  their 
product  as  stated  above.  But  as  just  shown,  that  is  impossible 
because  the  degree  of  the  result  would  prevent  it  from  satisfying 
the  differential  equation  of  which  it  purports  to  be  a  solution. 

It  can  in  fact  be  readily  shown  that  no  exact  algebraic  solution 
of  this  differential  equation  is  possible  that  will  fit  the  case  of  a  slab 
resting  on  relatively  stiff  beams  at  the  edges  of  the  panels.  In 
equations  (9),  (10),  (11),  (12),  however,  a  novel  solution  is  obtained 
which  will  be  used  as  a  basis  for  approximate  equations  applying 
to  a  slab  resting  on  beams.  It  is  evident  since  side  beams  are  de- 
signed of  arbitrary  cross  sections  to  carry  the  slab,  that  their  deflec- 
tions which  depend  upon  their  design  as  to  stiffness  relative  and 
absolute  is  the  determining  factor  not  only  of  the  slab  deflections 
but  of  the  shears  and  steel  stresses  of  the  slab.  Beams  and  slab 
are  consequently  independent  members  of  the  combination  and  are 
not  readily  amendable  to  treatment  as  a  simple  system. 

The  general  partial  differential  equation  of  the  surface  of  the 
middle  layer  of  a  continuous  flat  slab  loaded  uniformly  is,  see  equa- 
tion (20)  Chapter  V. 

54z  d4z  d*z    _(l~K2)g 

5x4^     dx2dy2  +    dy*    '          El 

in  which  x  y  z  are  the  coordinates  of  the  deflected  surface,  q  the 
intensity  of  the  uniform  loading,  K  is  Poisson's  ratio,  E  is  Young's 
modulus  of  elasticity,  and  /  is  the  moment  of  inertia  per  unit  of 
width  of  vertical  cross  section  of  the  slab  in  any  plane  parallel  to  z 


MORE    GENERAL    SOLUTION  279 

In  deriving  (8)  it  is  assumed  that  during  the  small  flexure,  which 
occurs  by  reason  of  the  loading,  z  only  varies,  and  that  the  coordi- 
nates x  and  y  of  any  given  point  of  the  slab  remain  unchanged,  which 
assumption  undoubtedly  is  sufficiently  in  accordance  with  fact  for 
technical  purposes. 

A  somewhat  more  general  form   of  solution  of   (8)   than  that 
given  in  Chapter  V  may  be  written  as  follows: — 

24#7(c!+c2)  z  =  q  (1—  K2)[Cl  (x2—a2)2+c2  (y2—b2)2] (9) 

in  which  Ci  and  c2  are  any  arbitrary  constants  whatever.  That  (9) 
in  fact  satisfies  (8)  and  is  consequently  a  particular  solution  of  (8), 
may  be  readily  verified  by  trial. 

A  form  of  solution  less  general  than  (9)  is  the  following,  which 
involves  but  one  arbitrary  constant  n  in  place  of  the  two  found  in  (9). 


which  is  a  form  of  solution  especially  applicable  to  the  single  panels 
of  a  continuous  slab  divided  into  rectangular  panels  of  size  2ax26, 
where  the  origin  of  coordinates  is  at  the  point  occupied  by  the  center 
of  the  panel  before  deflection,  and  the  axes  of  x  and  y  are  parallel 
to  the  edges  2a  =  Ll  and  2b  =  L2  respectively. 

It  will  be  noticed  that  the  corners  of  the  panel,  x  =  a  and  y  =  b, 
are  fixed  points  of  zero  deflection,  whatever  may  be  the  load.  These 
consequently  are  points  of  support  of  the  panel  with  reference  to 
which  other  points  x  y  undergo  the  deflection  z. 

It  is  to  be  observed  that  solutions  (9)  and  (10)  differ  in  effect 
from  solution  (21)  Chapter  V,  in  introducing  into  the  solution  unit 
moments  of  inertia  which  are  not  the  same  for  x  as  for  y.  For  let 
the  solutions  (9)  and  (10)  be  written  in  the  following  form: 

2  - a2)V/1  +  (y2  - b2  )V/2 

which  is  identical  with  (9)  and  (10)  provided 

71  ==  (ci  +  c2)  7/2Cl  ==  (a2n  +  62n  )  7/262n 

72  =  (ci  +  c2)  7/2c2  =  (a211  +  62n  )  7/2a2n 

Hence  the  modified  moments  of  inertia  I\  and  72  of  unit  widths  of 
slab  perpendicular  respectively  to  x  and  y  have  the  ratio  to  each  other 


280  UNEQUAL    MOMENTS    OF    INERTIA 


-,      or      -  =  -<_       - 


(2) 
2         Cl       6211  7      2  I  /!     72 

in  which  1/7  is  the  mean  of  the  reciprocals  of  the  moments  of  inertia 
1 1,  and  1 2 

For  n>0  and  a>6  we  have  /i>/2,  and  the  slab  is  stiffer  length- 
wise than  crosswise.  This  decreases  the  deflections  on  the  long  side 
compared  with  those  on  the  short  side  more  and  more  the  larger 
n  becomes  in  such  wise  that  they  are  equal  in  case  n  =  2. 

By  sufficiently  increasing  n  the  case  may  be  treated  where  the 
stiffness  along  x  is  any  required  multiple  of  that  along  n.  Solutions 
(9),  (10),  (11)  then  all  refer  to  a  slab  which  in  case  n  is  positive  is 
stiffer  and  has  more  steel  along  x  per  unit  of  width  of  slab  than  along 
y,  a  case  which  is  inconceivable  in  a  homogeneous  plate  but  per- 
fectly realizable  in  a  slab,  especially  in  a  slab  with  two  way  rein- 
forcement. As  shown  by  equation  (13)  the  deflection  on  the  stiffer 
long  side  will  be  reduced  so  as  to  become  equal  to  that  on  the  short 
side  or  in  other  words  201  =  z02  when  the  stiffness  along  x  is  so  in- 
creased that  n  =  2  or  /i//2  =  fr4/a4;  whereas  in  the  mushroom 
slab  in  which  7X  =  72  and  n  =  0  we  find  the  ratio  of  the  deflections 
at  mid  span  to  be  202/z0i  =  b^/a*. 

Assume  that  the  longer  side  is  2a,  so  that  a>6;  and  designate 
the  deflection  at  mid  span  on  the  longer  edge  where  x  =  0  and  y  =  b  by 
201,  and  by  z02  the  deflection  at  mid  span  on  the  shorter  edge  where 
x  =  a  and  y  =  0  as  shown  in  the  diagram  Fig.  74  showing  a  plan  of 
a  panel.  Also  let  z0  be  the  deflection  at  the  panel  center  where 
x  =  0  =  y.  Then  in  case  /  has  the  same  value  at  all  points  of  the 
slab,  we  find, 


z    = 


-+a)  ,02 

24  E  I  (a2n+62n)  z01 


a4 


J0j 


> (13) 


It  will  be  noticed  that  in  case  of  square  panels,  where  a  =  b,  all 
values  of  n  lead  to  identically  the  same  results,  viz.:  those  already 
discussed  in  Chapter  V,  where  in  fact  the  case  of  n  =  Q  was  treated. 
It  was  there  applied  to  flat  slabs  devoid  of  stiffening  girders  other 
than  those  forming  part  of  the  slab  itself,  and  having  a  moment 
of  inertia  no  greater  than  the  rest  of  the  slab.  In  fact  the  moment 


EFFECTS    OF    UNEQUAL    MOMENTS    OF    INERTIA  281 

of  inertia  at  the  side  belts  was  taken  as  somewhat  less  than  its  mean 
value  in  those  parts  of  the  panel  subject  to  the  maximum  applied 
bending  moments.  This  case  of  (10)  where  n  =  0  has  been  shown 
by  detailed  tests  to  agree  well  with  such  a  flat  slab  for  panels  of  an 
oblateness  as  great  as  b  /a  =  0.75. 

In  case  n  =  0  the  deflection  at  the  mid  spans  of  the  longer  and 
shorter  sides  of  a  uniform  slab  without  stiffening  beams  are  as  the 
fourth  powers  of  those  sides,  so  that  for  b  /a  =  .75  the  deflection  on 
the  shorter  side  is  not  quite  one  third  that  on  the  longer  side,  while 
one  fourth  of  the  total  load  on  the  panel  is  theoretically  carried  to 
each  edge  by  the  shears,  regardless  of  the  relative  lengths  of  the 
edges.  Quite  an  important  portion  of  the  discussion  of  flat  slabs 
was  devoted  to  the  investigation  of  how  this  shear  is  distributed  in 
the  beamless  uniform  slab  without  producing  prohibitive  stresses. 
Now  it  appears  from  (11)  that  in  case  n  =  l  /2  the  deflections  at  the 
panels  edges  are  as  the  third  powers  of  their  lengths,  just  as  occurs 
in  beams  with  equal  moments  of  inertia  loaded  with  loads  of  the  same 
unit  intensity.  It  will  be  shown  later  that  in  case  n  =  1  /  2  the  shear 
at  the  edges  of  the  panel  is  the  same  per  unit  of  edge  thruout  both 
the  longer  and  shorter  sides. 

The  results  just  considered  as  well  as  those  for  intermediate 
cases  will  be  found  together  with  other  matter  in  Table  1,  page  284. 

The  discussion  of  the  theory  of  the  beamless  flat  slab,  already 
referred  to,  was  rendered  possible  by  introducing  into  the  formulas 
which  were  employed  such  values  of  the  moment  of  inertia  /  as 
were  shown  to  exist  in  the  various  parts  of  the  panel  by  reason  of 
the  amount  and  position  of  the  reinforcement,  on  the  assumption 
that  the  action  of  the  reinforcement  could  be  replaced  without 
noteworthy  error  by  a  uniform  sheet  of  metal  of  the  same  weight  as 
the  actual  reinforcement. 

But,  in  case  of  a  slab  with  stiffening  beams,  it  would  manifestly 
be  incorrect  to  assume  that  the  effective  moments  of  inertia  may  be 
taken  as  approximately  the  same  both  along  and  across  an  edge. 
Hence  along  edges  the  value  of  I  will  be  large  compared  with  /  else- 
where, and  along  the  edges  y=-\-b  the  value  of  (x2 — a2)2//i  will 
nearly  or  quite  vanish  by  reason  of  the  largeness  of  /i,  the  moment  of 
inertia  in  the  girder  along  the  edge;  and  similarly  along  the  edges 
x=+a  the  value  of  (y2 — b2)2/I2  will  also  vanish.  It  is  in  this 
manner  that  the  slab  becomes  nearly  level  along  the  panel  edges. 
As  previously  stated,  the  effect  of  this  stiffening  of  the  edges  will 
need  to  be  considered  and  allowed  for  in  obtaining  practical  formulas 
for  deflection  and  stress  in  case  of  a  slab  resting  on  side  beams. 


282 


FURTHER   DISCUSSION    OF   UNEQUAL   MOMENTS   OF    INERTIA 


If  we  compare  the  value  of  z0  in  (11)  in  case  n  =  0  with  ZQ  in  case 
n  =  2  we  find  the  latter  is  the  smaller  when  a>b.  As  will  be  seen 
from  (10),  all  sections  of  that  surface  made  by  vertical  planes  parallel 
to  an  edge  are  curves  thai  are  identical  in  shape,  and  such  that 
2o  =  Zoi+2o2  in  which  20i  and  zQ2  maybe  regarded  either  as  the  mid 
deflections  of  the  edges  or  the  mid  deflections  of  the  meridian  curves 
of  the  surface  made  by  vertical  planes  respectively  parallel  to  the 
edges. 

Now  as  n  increases  from  0  to  2,  the  deflection  z0  decreases  some- 
what, but  the  deflection  20i  of  the  longer  side  decreases  more  rapidly 
than  ZQ,  while  the  deflection  z02  of  the  shorter  side,  or  the  crosswise 
deflection  the  short  way  of  the  slab  actually  increases.  This  appears 
from  inspection  of  Tables  1  and  2.  This  may  be  regarded  as  due 
to  the  increase  of  the  portion  of  the  total  load  carried  by  the  shears 
to  the  longer  edges,  so  that  the  loading  of  the  crosswise  reinforce- 
ment increases  with  the  increase  of  stiffness  along  x. 


It  appears  from  (10)  that 
5z  =  q  (l_-  -_J£2j>  62n 
5x~         6#/a2n 


—  a2x) 


by 


_ 
2n         2n 


GEI(an  +  6n) 


(14) 


Hence      —  =  0  at  x  =  0,  and  at  x  =  +  a 
ox 

and         -  =  0  at  y  =  0,    and  at  -y  =  +  b 
by 

consequently  the  panel  is  horizontal   across  its  meridian  sections 
and  across  its  edges,  regardless  of  n. 


Again 


52z       q  (1  -  K2)  62n  (3x2  -  a2)    ] 


5x2 


QE  I 


+ 


= 

b2y 


b2) 


.   ..(15) 


Also 


d2z 


bxdy  bx    by 


ELONGATIONS    AND    TWISTING    MOMENTS  283 

Let  /  =  ij  d2  A  in  which     I/A  ==  1/2  (l/A1  +  1/A2)  where  I/A 
is  the  mean  of  the  reciprocals  of  the  steel  areas 

,S2z        q  (1  —  K2)  62n  (3x2  —  a2)   1 
Hence    e\  =  +  i  d—-n  =   J 


2  ___?r_? 

and        6=  +i          =  l^ 


.(16) 
(of 


Note  that  e\  is  independent  of  y,  and  e2  independent  of  x. 
Designate  the  unit  elongation  along  x  at  x  =  0  by  e01,  and  along 
y  at  0  =  0  by  602,  then  e02/e0i  ==  (6/a)2~2n (17) 

In  case  n  =  Q,  the  elongations  (and  unit  stresses)  in  the  reinforc- 
ing rods  are  proportional  to  the  squares  of  their  lengths,  but  in  case 
n  =  2  they  are  inversely  as  the  square  of  their  lengths,  and  the  short 
cross  rods  are  under  the  greater  stresses.  Other  intermediate  cases 
are  shown  in  Table  1. 

It  will  be  noticed  that  the  signs  of  the  e-i  and  e2  change  at  lines  of 
inflection  situated  at  the  same  positions  as  in  the  mushroom  slab 
giving  negative  bending  moments  across  the  edges  of  the  panels 
twice  as  large  as  the  positive  moments  across  the  meridian  lines 
parallel  to  those  edges.  These  negative  moments  do  not  in  general 
require  any  reinforcement  at  the  top  of  the  slab  across  the  panel 
edges  because  they  are  resisted  sufficiently  by  the  reinforcement 
running  perpendicular  to  these  edges,  to  which  the  entire  negative 
moments  are  transferred  laterally  by  twisting  moments  induced  in 
the  panel.  This  is  the  same  kind  of  action  that  occurs  in  the  Mush- 
room slab,  by  which  the  applied  negative  moments  across  the  panel 
edges  are  carried  laterally  by  twisting  moments  toward  the  columns, 
until  they  are  held  in  equilibrium  by  the  side  belts  running  at  the 
top  of  the  slab  over  the  columns. 

The  twisting  moments  here  mentioned  are  not  to  be  derived 
analytically  from  (3)  because  that  equation  contemplates  the  case 
of  reinforcement  distributed  thruout  the  panel  to  resist  negative 
as  well  as  positive  moments  wherever  they  may  occur.  In  case  of 
distributed  reinforcement  the  magnitude  of  the  twisting  moment  in 
any  vertical  plane  parallel  to  the  edges  per  unit  of  width  of  slab  is 

El.       dz 

n  =-  (18) 

1  +  K         d  x  by 


284 


SHEARING    STRESSES 


therefore  n  =  0  by  (17),  and  the  only  twisting  moment  in  vertical 
planes  parallel  to  the  edges  is  that  due  to  the  action  just  stated,  viz. 
that  the  steel  required  to  resist  negative  moments  is  not  to  be  found 
distributed  across  the  panel  edges  but  instead  is  concentrated  in 
parallel  positions  at  the  edges  of  the  panel.  Such  twisting  moments 
induce  shearing  stresses  in  steel  and  concrete  in  vertical  planes 
parallel  to  the  edges,  but  in  amounts  and  with  a  distribution  such 
as  not  to  require  investigation  here. 

The  intensities  of  the  shearing  stresses  in  (10)  per  unit  of  width 
of  slab  found  by  equations  (9)  and  (15)  Chapter  Y,  are 


E  I 


5  /  82z 


-s2  = 


5x 


dy 


/  52z  S2*  \         q  b2nx     N 

\  bx2  "  K  by2)     a2n+b2n 

82z    \         q  a^y 

TS*>        ?    \    ?    2  ~T   ^    o    2      I  =     2n     i     i.2n 

— A"     oy\oy  ox    /      a     -\-  b       ' 


-  K'z     5x\  bx2 
El 


>....(19) 


Hence  the  shear  at  any  distance  x  from  the  center  is  independent 
of  y,  and  vice  versa.     The  total  shears  at  the  edges  x  =  a  and  y  =  b  are : 


on  edge  26,  --  2  b  Sl  =  2  q  a  62n+1/(a2n  +  62n) 
on  edge  2a,   -  -  2  a  s2  =  2  q  a2n+1  6/(a2n  +  62n) 
S2/Sl  =  (b/a)1-211,   as2/bSl  =  (b/a)~2n 


(20) 


This  distribution  of  shear  on  the  edges  when  n  >  1  is  very  dif- 
ferent from  that  occurring  in  the  uniform  slab  supported  on  columns 
where  n  =  0.  The  total  shear  on  the  four  edges  is  in  any  case 
twice  the  sum  of  the  shears  on  one  short  and  one  long  edge  as  just 
obtained,  and  amounts  to  W  =  4  q  a  6,  the  total  load  on  the  panel. 

TABLE  1. 


n       Q 

1/2 

1 

3/2 

2 

^02/^01  •  •  •  •  (b/a)2 
s-2/Si  b/a 

b/a 

1 
(6/a)-1 

(b/a)2 
1 
(6/a)-1 
(6/a)-2 

b/a 

(b/a)-1 
(b/a)-2^ 

1 

(6/a)-2 

RECTANGULAR  PANELS  RESTING  ON  BEAMS 


285 


I* 


**«« 


The  diagram  in  Fig.  74  shows  a  plan  of  a  single  panel  arranged 
to  show  the  notation  for  deflections,  shears  and  elongations  in  con- 
nection with  the  formu- 
las  of  this  paper.  Table 
1  expresses  the  ratios  of 
these  quantities  for  vari- 
ous values  of  n  between 
0  and  2  inclusive,  and 
Table  2  gives  numerical 
values  of  such  ratios  in 
case  b/a  =  0.75.  From 
these  the  truth  of  pre- 
vious statements  as  to 
the  relative  magnitudes 
of  various  deflections,  shears,  elongations,  etc.,  will  be  apparent. 

TABLE  2. 


-X- 


I* 


Fig.  74.     Diagram  of  Notation. 


n 

0 

1/2 

1 

3/2 

2 

wv 

0.32 

0.42 

0.56 

0.75 

1 

202/20-   •      •  : 

0.24 

0.3 

0.36 

0.43 

0.50 

zoiAo- 

0.76 

0.7 

0.64 

0.57 

0.50 

002/001-     •  •  • 

0.56 

0.75 

1 

1.33 

1.77 

82/81  

0.75 

1 

1.33 

1.77 

2.37 

as2/asi  .... 

1 

1.33 

1.77 

2.37 

3.16 

5.     Slab  with  Rectangular  Panels  Supported  on  Beams.     It  is 

usual  practice  for  architectural  and  other  reasons  to  make  the  beams 
on  the  long  and  short  sides  of  panels  of  the  same  depth.  In  case 
the  slab  is  uniformly  loaded  thruout  assume  that  the  steel  will  have 
equal  unit  stresses  in  both  beams.  In  order  that  this  may  occur  it 
is  necessary  to  know  the  loads  that  are  transmitted  to  the  beams 
from  the  slab. 

It  may  be  shown  that  the  load  which  comes  upon  the  beams  is 
nearly  uniformly  distributed  when  the  slab  is  uniformly  loaded  thru- 
out  and  that  the  load  per  unit  of  length  is  nearly  the  same  for  both 
side  beams  even  for  considerable  variations  of  the  relative  stiffness 
of  the  side  beams. 


286  '      SIDE    BEAMS 

While  the  same  relative  distribution  of  load  would  continue  to 
hold  in  case  of  the  beams  at  the  edges  of  a  single  loaded  panel,  the 
total  loads  upon  these  beams  would  be  reduced  to  one  half  those 
of  a  slab  loaded  uniformly  thruout.  But  the  twisting  of  the  beams 
due  to  the  unbalanced  load  on  one  side  of  the  beam  would  induce 
unequal  stresses  in  the  several  reinforcing  rods  of  the  beam  so  that 
the  stresses  in  the  rods  next  to  the  loaded  area  might  experience 
little  or  no  reduction  of  stress  from  removal  of  load  from  all  except 
one  panel. 

In  case  of  three  adjacent  panels  loaded  with  their  long  sides  in 
common,  the  reduction  of  the  deflection  at  their  ends  or  short  sides 
to  about  one  half  of  that  in  case  of  uniform  loading  thruout  will 
have  some  effect  to  increase  the  stresses  in  the  longitudinal  rein- 
forcement to  the  relief  of  the  crosswise  reinforcement,  which  latter 
will  be  shown  later  to  be  under  the  more  severe  stresses.  This  case 
therefore  does  not  need  special  consideration,  and  the  case  of  uniform 
load  thruout  alone  needs  be  provided  for. 

The  fact  that  in  case  of  full  load  on  the  slab  all  the  side  beams  have 
practically  the  same  unit  load  is  due  to  the  peculiar  action  of  the 
beams  in  producing  a  kind  of  flexure  in  the  slab  which  is  very  different 
in  its  nature  from  what  occurs  when  it  is  supported  on  columns. 

In  case  of  column  support  and  stiff  heads,  the  surface  of  the  slab 
which  is  convex  upward  about  the  column  center  has  a  uniformity 
of  curvature  that  ensures  cantilever  action  and  concentric  circum- 
ferential stresses  which  are  practically  uniform  completely  around 
the  column.  If  any  slight  deviation  from  this  actually  occurs  it  may 
be  assumed  to  be  represented  by  a  slightly  greater  extension  of  the 
cantilever  area  along  the  diagonals  than  along  the  sides,  but  the 
difference  is  practically  negligible. 

With  deep  side  beams  and  a  comparatively  flexible  slab  all  this 
is  changed  under  heavy  loads.  Large  parts  of  what  in  case  of  no 
side  beams  would  be  cantilever  area  is  changed  by  the  introduction 
of  beams  into  hollow  valleys  running  up  toward  the  column  centers 
at  an  angle  of  45  degrees  with  the  beams.  The  bending  moments 
across  these  valleys  which  in  the  simple  cantilever  slab  were  nega- 
tive have  changed  sign,  and  this  change  has  profoundly  modified 
the  mechanical  action  of  the  slab.  The  surface  about  the  column 
center  instead  of  being  practically  uniformly  convex  now  has  four 
valleys  and  four  ridges  radiating  from  the  column,  and  it  is  evident 
as  previously  stated  that  no  expression  is  possible  which  is  of  alge- 
braic form  merely,  that  will  express  such  relations.  It  would  require 


VALLEY    LINES  287 

certain  trigonometric  expressions  of  multiple  angle  about  the  column 
center  to  express  this  scallop-shaped  surface. 

At  the  diagonal  center  of  the  panels,  however,  conditions  are 
sufficiently  unchanged  by  the  side  beams  to  admit  of  approximate 
algebraic  expression  of  the  mechanical  and  geometrical  relations. 

The  condition  of  the  slab  may  be  regarded  as  having  been  brought 
about  from  the  initial  condition  of  a  uniform  slab  supported  on 
columns  with  the  usual  side  and  center  deflections  by  the  deforma- 
tions which  would  be  produced  in  it  by  stiffening  or  jacking  up  the 
sides  sufficiently  to  reduce  their  deflections  by  two  thirds  or  three 
quarters  of  their  initial  amounts.  This  would  bend  the  edges  up- 
ward enough  to  form  the  valleys  before  mentioned,  and  at  the  same 
time  greatly  reduce  the  width  of  the  saddle  across  each  side.  These 
reductions  make  radical  changes  in  the  nature  of  all  the  curvatures 
near  the  sides  which,  as  has  been  said,  cannot  be  readily  expressed 
algebraically. 

But  certain  aspects  of  these  phenomena  admit  of  satisfactory 
graphical  expression.  It  is  known  from  a  considerable  body  of 
experimental  work  on  slabs  with  wall  supports  or  deep  beams  at  the 
panel  edges  that  for  heavy  loads  the  valley  lines  will  occur  at  angles 
of  practically  45  degrees  with  the  sides  regardless  of  the  relative 
lengths  of  the  sides  or  of  the  continuity  of  the  panels.  The  valley 
lines  are  of  necessity  lines  of  maximum  moments  positive  across  them 
and  consequently  define  at  the  same  time  the  lines  of  zero  shear. 
The  loads  that  go  to  the  sides  may  consequently  be  computed 
approximately  from  consideration  of  these  lines  across  which  no 
loads  are  transmitted. 

Draw  lines  from  the  ends  of  the  short  sides  of  the  panel  at  angles 
of  45  degrees  with  them  thus  forming  two  right  angled  isosceles 
triangles.  The  apex  of  each  of  these  triangles  lies  on  the  meridian 
section  of  the  panel  made  by  a  vertical  plane  midway  between  the 
long  sides  of  the  panel.  The  areas  of  these  two  triangles  and  the 
two  halves  of  the  remaining  area  of  the  panel  lying  on  either  side 
of  the  meridian  section  consequently  show  approximately  the  rela- 
tive loads  carried  to  the  beams  on  the  short  and  long  sides  of  the 
panel.  The  load  on  each  short  side  beam  will  be  W  b/2a  and  on 
each  long  beam  W  [b2  +  2  (a  —  6)  6]/2  a  b  giving  us  a  total  load  of 
2  W  on  the  four  side  beams  of  which  W  comes  from  the  panel  itself 
and  W  from  the  surrounding  panels.  This  distribution  of  loads  may 
be  assumed  to  be  exact  in  case  of  rigid  wall  supports  and  approximate 


288 


LOADS    ON    SIDE    BEAMS 


for  stiff  beams.  The  relative  values  of  these  quantities  for  vary- 
ing proportions  of  sides  are  shown  graphically  in  the  diagram  Fig. 
75.  Fig.  75  also  shows  the  relative  values  on  the  assumption  that 
the  panel  loads  are  uniformly  distributed  along  the  four  sides  as 
expressed  in  equation  (21)  given  later. 


00 


Diagram  of  Loads  on  Side  Beams  on  Various  Assumptions 
Fig.   75 

The  general  agreement  of  these  two  assumptions  with  each  other 
is  sufficient  to  enable  us  to  adopt  a  uniform  distribution  on  the 
perimeter  as  a  convenient  and  closely  correct  basis  of  distribution. 
It  might,  however,  at  first  be  thot  that  the  loads  would  be  more 
concentrated  toward  the  middle  of  the  sides,  but  the  twisting  mom- 
ents such  as  have  already  been  discussed  in  case  of  mushroom  slabs 
largely  prevents  such  inequality.  These  twisting  moments  transfer 
the  applied  negative  moments  along  the  sides  even  tho  there  is  little 
or  no  reinforcement  in  the  top  of  the  slab  across  the  beams  to  resist 
them  and  apply  them  to  the  side  beams  at  the  columns,  which  beams 
are  so  stiff  that  it  seems  to  be  of  little  importance  whether  the 
reinforcement  which  is  carried  across  the  side  beams  is  at  the  top 
surface  or  not.  The  exact  distribution,  however,  is  dependent  to 
some  extent  upon  the  relative  stiffness  of  the  side  beams.  The 
assumption  of  a  uniform  distribution  is,  however,  sufficiently  exact 
for  practical  purposes. 


COMMON    ASSUMPTION    ERRONEOUS  289 

It  would  seem  appropriate  at  this  point  to  refer  to  another  pro- 
posed distribution  of  loading  which  was  put  forth  by  Marsh  in  his 
Reinforced  Concrete,  pages  282  et  seq.  where  it  was  attempted  to 
l)e  shown  on  the  basis  of  beam  strip  theory  that  the  loads  on  the 
sides  are  proportional  to  the  fourth  powers  of  the  sides.  This 
result  has  been  incorporated  into  building  codes  and  text-books 
almost  universally.  But  in  the  opinion  of  the  writers  has  not  a 
scintilla  of  evidence  to  support  it  either  in  correct  theory  or  experi- 
ment. 

Any  correct  theory  would  have  to  find  a  separate  equation  for 
the  curvatures  of  the  parts  of  the  panel  into  which  it  is  separated 
by  the  valleys  which  are  known  to  exist.  These  valleys  are  of  such 
a  nature  as  to  prevent  continuity  of  beam  action  such  as  was  assumed 
to  exist  in  order  to  arrive  at  these  erroneous  conclusions.  Since  it 
has  been  sufficiently  shown  that  no  algebraic  expression  is  possible 
which  will  exactly  express  the  relationship  here  existing  it  is  clear 
that  were  this  result  correct  for  the  case  of  supporting  walls  it  could 
hardly  hold  at  the  same  time  for  supporting  beams  also.  The  wide 
divergence  of  this  theory  from  the  previous  nearly  concordant  esti- 
mates appears  from  the  diagram  in  Fig.  75  where  the  relative  beam 
loads  on  this  theory  are  shown  in  a  manner  comparable  with  the 
uniform  distribution  here  adopted.  Such  a  theory  would  render 
longitudinal  reinforcement  practically  useless  in  any  slab  whose 
width  is  more  than  a  few  percent  less  than  its  length.  The  inherent 
improbability  of  this  may  be  regarded  as  a  sufficient  disproof  of  this 
so  called  law. 

This  is  evident,  because  the  fact  of  the  existence  of  45°  valleys 
at  which  maximum  moments  exist  seems  in  itself  to  make  it  certain 
that  the  steel  stresses  diagonally  across  these  valleys,  or  in  other 
words  the  stresses  parallel  to  the  long  and  short  sides  of  the  panel 
must  be  equal.  This  would  require  practically  the  same  reinforce- 
ment per  unit  of  width  of  slab  lengthwise  as  crosswise  of  the  panel 
at  the  valleys  where  the  stresses  may  be  regarded  as  critical.  It 
would  thus  appear  that  the  reinforcement  instead  of  being  largely 
superfluous  longitudinally  should  be  practically  of  the  same  amount 
per  unit  of  width  lengthwise  and  crosswise  of  the  slab,  a  require- 
ment in  the  most  striking  contrast  with  the  common  but  erroneous 
theory  just  mentioned.  As  the  loading  of  such  a  slab  becomes  more 
severe  and  the  point  of  failure  is  approached  the  stresses  at  the  valleys 
apparently  increase  more  rapidly  than  elsewhere  and  the  phenomena 
accompanying  them  become  more  pronounced.  This  ultimate 


290  DEFLECTIONS    OF    SIDE    BEAMS 

condition  is  the  controlling  condition  and  due  provision  for  it  is 
essential  in  correct  design. 

If  the  unit  shear  at  the  edges  of  the  panel  be  taken  as  constant 
and  the  width  of  the  beams  be  disregarded  then  the  load  that  is 
supported  on  each  unit  of  length  of  a  side  beam  of  a  panel  in  a  slab 
which  is  uniformly  loaded  thruout  with  a  total  load  of  W  per  panel 
will  be  twice  as  much  as  comes  to  that  side  from  the  panel  itself. 
The  total  perimeter  is  4  (a  +  b)  and  the  total  load  on  the  sides  is 
2W.  Hence  w  =  W/2  (a  +  6)  is  the  load  per  unit  on  the  side 
beams. 

The  total  load  on  long  side=  2aw  =Wa  =  Wa/(a  +  6)1 
"       "         "      "  short     "    =  2bw  =Wb  =  Wb/(a  +  6)J 

The  deflection  formulas  for  these  continuous  beams  will  be 

24  E  Iaza  =  w  (x2  —  a2)2  =  W  (x2  —  a2)2/2  (a  +  6)1 

24  E  Ibz2  =  w  (y2--b2)2   =  W  (y2  --  62)2/2  (a  +  6)J(22) 

Let  7a  and  7b  denote  the  moments  of  inertia  of  the  long  and  short 
side  beams  respectively.  Let  these  side  beams  be  of  equal  depth 
as  usually  designed  and  let  them  have  the  same  unit  stress  in  the 

steel.     Then  7a//b  =  Aa/Ab  =  a2/b2 (23) 

since  the  moments  of  resistance  are  proportional  in  that  case  to  the 
cross  sections  Aa  and  Ab  of  the  reinforcements  and  the  applied  mo- 
ments at  mid  span  are  proportional  to  the  squares  of  the  spans. 

By  (23)  ---      = 


Hence  za   = 


W  (a2  +  b2)  (x2  -  a 


22 


48  E  (7a  +  /b)  a'  (a  +  6) 

. . (25) 

48  E  (Ia  +  /b)  62  (a  +  6)  ^ 

These  equations  permit  us  to  compare  the  deflections  at  mid 
span  zoa  and  zob  with  each  other  and  with  the  deflections  that  occur 
in  mushroom  slabs  at  the  same  points. 

2ob/ zoa    =    b  /a 

In  case,  b/a  =  0.75,  we  have  zob/zoa  =  0.565. 

Compare  the  deflection  at  mid  span  of  the  long  side  beam  zoa 
with  the  mid  span  deflection  zob  in  the  long  side  belt  of  a  Mushroom 
slab  having  the  same  total  load  per  panel  and  the  same  amount  of 


COMPARED    WITH    MUSHROOM    SLAB  291 

steel  in  its  four  belts  as  in  the  two  side  beams,  that  is  Aa  +  Ab  =  4A^ 
By  (30)  Chapter  V,  the  deflection  at  mid  span  of  the  long  side  of 
the  mushroom  panel  is 

(1  —  K2)  W  a3 


48  E  ijd21QA1 

By  (25)  above  the  deflection  at  mid  span  of  the  long  side  beam  is 
W  (a2  +  b2)  a2 


a)      (dV  (26) 

2      i      j^    jl  0    /-,       ,      ^2   /    2x    ^  ^ 


(\—K2)a(a  +  b)d2  (1  +  &/ 

1.5  (a2  +  b2)  dl  2  (1.+  6 

If  d/d1  =  2.5,  (d/dj2  =  6.25 

"     dA  =    3,  (d/dO2  =  9. 

"  b/a  =  .75,  then  z01/z0&  =  0.56  (d/dj2 

In  case  (d/di)  =  2.5,  then   zol/z0&  =  3.5. 
=  3,        «       ^Ol/^oa  -  5. 


from  which  it  appears  that  the  deflection  of  the  side  beam  is  from 
one  third  to  one-fifth  that  of  a  mushroom  slab,  dependent  upon 
relative  depths  of  slab  and  beam. 

It  should  be  noticed  that  reinforcing  steel  of  the  slab  which  is 
near  by  and  parallel  to  either  side  beam  lies  at  a  level  compared 
with  that  in  the  beams  at  which  it  is  able  to  offer  resistance  to  the 
negative  moments  in  the  beams  near  the  columns.  In  particular 
it  largely  prevents  the  propagation  of  moments  across  the  column 
heads  due  to  unbalanced  loads  such  as  occur  with  single  or  alternate 
panels  loaded.  It  therefore  assists  that  part  of  the  beam  reinforce- 
ment which  is  near  the  top  of  the  slab  over  the  columns. 

Having  treated  the  side  beams  consider  now  the  deflections  of 
the  slab  supported  on  side  beams.  It  is  evident  from  the  preceding 
discussions  that  while  the  curvatures  of  the  meridian  or  central 
sections  of  the  slab  by  vertical  planes  parallel  to  the  sides  are  greatly 
increased  near  the  sides  the  curvatures  of  these  sections  near  the 
panels  centers  are  not  so  much  changed.  The  changes  which  do 
occur,  however,  are  such  as  increase  the  curvature  of  the  cross- 
wise section  and  flatten  that  part  of  the  lengthwise  meridian  section 
which  lies  between  the  apices  of  the  valleys.  This  is  equivalent  to 


292  COMPUTED    DEFLECTION    D 

increasing  the  ratio  202/Zoi>  in  (11)  if  we  suppose  that  the  same  form 
of  (10)  will  approximately  represent  the  actual  surface  for  the  cen- 
tral portion  of  the  slab.  It  is  evident  that  the  surface  n  =  2  will 
not  make  sufficient  allowance  for  the  effect  of  the  valleys,  because 
with  n  =  2  the  surface  would  have  the  deflections  at  midspans  of 
the  long  and  short  sides  equal,  whereas  the  actual  surface  has  a  deeper 
crosswise  deflection  than  this  by  reason  of  reduction  of  its  saddles 
and  also  has  a  flatter  central  portion  between  valley  apices. 

Designate  the  deflection  of  the  panel  center  below  the  mid  span 
of  the  long  side  beam  by  D  and  assume,  since  z02  from  (13)  is  not  the 
total  deflection  of  the  center  below  mid  span  of  the  long  beam,  that 
an  approximate  value  may  be  obtained  by  increasing  this  value  of  z02 
in  the  ratio  of  the  sides  a/b. 

Assume  that  the  central  deflection  z02  for  n  =  2  will  be  increased 
in  this  manner  to 

(1  —  K2)  q  a5  b3 


an  assumption  that  will  need  to  be  verified  by  experiment  as  it  seems 
in  fact  to  be  so  verified. 

This  approximate  expression  for  the  deflection  D  is  intended  to 
express  the  difference  of  level  between  the  mid  span  of  the  long  side 
beam  and  the  panel  center.  It  is  not  thot  possible,  however,  to  ob- 
tain any  closely  approximate  expression  for  the  difference  of  level 
between  the  mid  span  of  the  short  side  beam  and  the  panel  center 
because  of  the  great  discontinuities  of  curvature  that  occur  at  the 
apices  or  points  of  intersection  of  the  valleys. 

In  order  to  obtain  a  practical  and  convenient  form  of  this  proposed 
deflection  formula  in  which  the  percentage  of  reinforcement  parallel 
to  each  side  is  assumed  to  be  the  same  let 

I  =  ij  d2  A  ......................................  (28) 

in  which  A  is  the  cross  section  of  one  unit  of  width  of  a  uniform 
sheet  of  metal  whose  weight  is  equal  to  that  of  the  reinforcing  rods. 

Take  the  case  of  two  way  reinforcement  parallel  to  the  edges  of 
the  panel 
Let      'ZA  !=  the  total  across  section  of  all  the  rods  running  the 

long  way  of  the  panel. 
and      2A2=  the  cross  section  of  these  running  the  short  way  across 

the  panel. 
then     ZA  8=    2^-f  2  A  2=  the  total  right  cross  section  of  slab  steel 

in  square  inches. 


VERIFIED    BY    OBSERVATIONS 


293 


Let         A0=  mean  right  cross  section  of  slab  steel  per  unit  of  width 
of  each  single  belt. 

then  AQL2  =  2L4.1,  and  ^o^i  =    2A2. 
Hence  A0  (Li  +  L2)  =   2  A 8,  and  Ll  L2  =    area  of  panel. 
Again  40  L!  L2  =    total  volume  of  steel  in  each  belt, 
and   V  =  2AQ  LI  L2  =    total  volume  of  both  belts. 

Therefore   V  /Li\  L2  =  2A0  =    thickness  of  equivalent  uniform 
sheet. 

But  by  definition  A  =  V/L1  L2,  hence  A  =  2A0, 
Hence  2AS  =  J  A  (Lx  +  L2), 

2  2  A*  2AK 


or   A  = 


a  +  b 


.(29) 


Substitute  (29)  in  (28),  and  (28)  in  (27), 

and  put  4  q  a  b  =  W,  K  =  0.5,  i  =  2/3,  j  =  0.89,  E  =  3  x  107,  and 
(1  +64/a4)/(l+6/a)  =  b/a.  This  last  is  an  approximate  numerical 
value,  true  for  b/a=  1,  and  6/a  =  0.75.  We  then  have  the  practical 
deflection  formula 

1     or»          i  /-ilO     i2  -^      *        \y^J 


an  expression  in  which  the  numerator  may  also  be  written  W  C2  L3, 
where  C2  =  L2/L±.     The  deflections  D  consequently  vary  directly 

as  C2  =  L2/Li  =  b/a. 

The  empirical  formula  in  Turner's  Concrete  Steel  Construction 
pages  55  and  56  and  found  also  on  page  62  above  gives  practically 
the  same  deflection  as  (30)  in  square  panels,  and  slightly  greater  de- 
flections for  a>6. 

Computed  deflections  may  be  compared  with  the  following- 
test  data  1,  2,  and  3,  taken  from  Turner's  Concrete  Steel  Construc- 
tion, pages  56  and  57: 

Deflections 


Building 

Computed  by  (8) 

Observed 

1  Minneapolis  Paper  Co  
2  Smythe  Block  

0.298" 
0  1148" 

0.30" 
0  11" 

3  Minneapolis  Knitting  Co  
4  Minneapolis  Armory  
5  Nicollet  Associates  Bldg  

0.1696 
0.657 
0.2497 

0.167 
0.75 
0.25 

294  GREATEST    STEEL    STRESS 

The  Minneapolis  Armory  panel  had  one  edge  on  a  wall  with  a 
wall  above  and  three  edges  on  beams  with  the  steel  raised  slightly; 
panels  20'  by  20'  from  center  to  center  of  girders  and  19'  clear  spans. 
Thickness  5.25"  at  center,  6.5"  at  edge.  Reinforcement  f  inch 
rounds  at  9"  between  centers.  Load  40  pounds  per  square  foot. 
Observed  deflection  f  ". 

160,000  (240)3 

=  U.bo/ 


1.82  X  10     X  4.25  X  4.25  X  52  X  .196 

The  discrepancy  in  case  of  this  panel  is  due  to  several  causes: 
It  was  not  built  into  the  Avail  it  rested  on.  Unusually  large  varia- 
tions of  thickness  occurred  in  it.  By  reason  of  scant  thickness  it 
was  over  reinforced  and  stresses  in  concrete  were  excessive. 

The  Nicollet  Associates  Building:  Panels  20'  5.5"x24'  2.5".  Thick- 
ness of  rough  slab  7"  and  1.75"  strip  fill.  Mean  thickness  7|  inches. 
Reinforcement  7/16"  rounds,  hard  grade  high  carbon  steel,  7" 
center  to  center  in  the  middle  third  of  spans,  and  9"  center  to  center 
in  the  rest  of  the  span.  Load  200  pounds  per  square  foot.  De- 
flection J  inch. 

D=  99058  x  245.5  (290.5)5  24Q7// 

1.82  x  1010  x  6.72  x  6.72  x  .15  x  67 

We  consequently  feel  justified  in  asserting  that  (30)  is  in  good 
agreement  with  experimental  data. 

Now  assume  that  by  reason  of  the  increased  curvature  the 
greatest  elongation  e02  at  the  panel  center  is  increased  in  the  same 
ratio  a/6  as  the  center  deflection.  Then  the  greatest  unit  stress 
in  the  cross  reinforcement  at  the  center  derived  by  taking  n  =  2  will 
be  given  by  the  expression 


F, 

=  E  €02  a 


/h 
b  = 


6  j  d  (a    +  o  )  LA 


W  Ll/L2      WLi/C2 
or    /vjo  = 

57  d2A«       57 


> (31) 


provided  the  same  substitutions  be  used  in  deriving  this  final  prac- 
tical formula  as  were  employed  in  obtaining  (30) . 

The  rsteel  stresses  obtain  by   (31)   vary  as  l/C2  =  Ll/L2  =  a /b. 


MINIMUM    SLAB    THICKNESS  295 

Applied  to  the  buildings  previously  mentioned  in  connection 
with  measured  deflections,  (31)  would  give 

Building  /s2  Computed  by  (31) 

Minneapolis  Paper  Co 22500  Ibs.  per  sq.  in. 

Smythe  Block 4287  "      "         " 

Minneapolis  Knitting  Co 9190  "      "         " 

Minneapolis  Armory 15500  "      "         " 

Nicollet  Associates  Bldg 8890  "      " 

The  stresses  computed  by  (31)  are  somewhat  smaller  than  those 
given  by  Turner's  empirical  formula  for  safe  design  in  his  Concrete 
Steel  Construction  page  55,  and  also  found  on  page  62  above. 

It  is  not  certain,  however,  that  the  stresses  across  the  line  joining 
the  apices  as  above  computed  are  greater  than  or  even  so  great  as 
those  across  the  valleys  at  points  near  the  apices.  It  is  believed, 
however,  that  any  larger  stresses  at  such  points  will  by  reason  of 
their  concentration  yield  sufficiently  to  bring  into  play  nearby  rods 
in  a  way  that  will  afford  relief  from  any  dangerous  stresses. 

Without  having  been  able  to  make  exhaustive  tests  sufficient  to 
completely  establish  the  practical  accuracy  of  the  theoretical  evalua- 
tion of  stresses  as  here  proposed  the  writers  nevertheless  have  highly 
corroborative  experimental  evidence  in  support  of  the  substantial 
correctness  of  equation  (31)  for  stresses. 

6.  Steel  Ratios  and  Minimum  Thickness  of  Slab.  Since 
slabs  should  be  so  designed  that  the  steel  would  yield  before  the 
concrete,  it  is  important  to  determine  how  large  a  percentage  of 
steel  may  be  introduced  without  passing  this  limit.  This,  however, 
is  dependent  upon  the  relative  thickness  of  the  slab.  In  case  of  the 
continuous  slab,  the  ratio  of  the  depth  to  side  span  involves  some- 
what different  considerations  from  that  of  the  ordinary  beam  since 
the  minimum  thickness  of  the  slab  on  a  diagonal  span  is  relatively 
less  than  is  permissible  in  beam  construction.  That  part  of  the 
thickness  of  the  slab  which  serves  as  fireproofing  and  which  is  con- 
stant for  any  given  size  of  rods  bears  a  greater  ratio  to  the  depth 
than  is  the  case  in  beam  construction.  Furthermore,  the  depth 
necessary  for  proper  embedment  over  the  support  and  the  massing 
of  the  steel  at  the  support  reduces  the  effective  lever  arm  of  the 
steel  in  the  cantilever  to  jsd,  which  reduces  it  to  a  relatively  greater 
extent  than  at  mid  span  where  the  effective  depth  becomes  j^d  as 
it  does  in  the  case  of  one  way  slab  construction  also.  This  reduc- 
tion in  effective  depth  for  fireproofing,  embedment  and  massing  of 


296  STEEL  RATIOS  IN  SLABS 

the  steel  is,  however,  a  constant  for  a  given  size  and  arrangement 
of  reinforcement  and  for  the  continuous  slab  it  is  found  that  we 
may  accordingly  take  this  constant  as  follows: 

2  "  for  3/8"    rod  reinforcement  (or  smaller  sizes) 
21-"  "     7/16"    " 

1\"   "     1/2"      "  " 

3  "  "     7/8"      "  " 

Tine  limiting  minimum  thickness  of  slab  then  may  be  stated  as 
L/48  plus  the  constant  just  tabulated  for  the  various  sizes  of  rods. 

In  case  square  twisted  rods  are  used,  the  same  constant  should 
be  used  for  3/8"  square  twisted  as  for  1/2"  rounds,  and  for  1/2" 
square  twisted  the  same  constant  as  for  5/8"  rounds.  This  will 
give  us  a  satisfactory  minimum  thickness  of  slab  for  all  spans  of 
this  type,  L  being  the  longer  direct  distance  between  column  centers. 

For  square  panels  supported  on  beams  or  walls  at  the  edges,  the 
constant  for  the  various  size  of  reinforcing  rods  should  be  one  inch 
less  than  that  tabulated  above,  and  the  minimum  thickness  taken 
as  L/48  plus  this  new  constant. 

Where  the  panel  is  rectangular  and  not  square,  and  supported  on 
the  sides  by  beams,  the  same  rules  will  hold  by  using  2  (az-\-b2)/(a-\-b) 
in  place  of  L  for  the  span  of  the  square  panel,  in  which  a  and  6  are 
the  half  spans  in  the  two  directions  respectively. 

Determination  of  reinforcement  required  in  slabs.  In  case  of  beams 
with  different  values  of  N  the  ratio  of  depth  to  span,  the  proper 
values  of  the  steel  ratio  p  were  obtained  in  Chapter  III  page  86. 
In  case  of  slabs  the  ratios  so  obtained  should  not  be  exceeded,  but 
in  reckoning  the  reinforcement  of  slabs  account  must  be  taken  of 
the  fact  that  the  steel  is  in  multiple  directions  so  that  the  reinforce- 
ment per  belt  will  be  only  a  fraction  of  the  total  permissible  steel. 

In  the  continuous  slab  with  four  way  reinforcement  and  Mush- 
room heads,  having  given  the  ratio  of  the  thickness  to  span  taken 
center  to  center  of  columns,  determine  the  percentage  of  steel  from 
the  diagram  for  beams,  page  86,  and  divide  this  by  2  to  find  the  limit- 
ing percentage  of  steel  for  each  belt. 

In  the  slab  with  two  way  reinforcement,  supported  on  columns, 
the  side  belt  may  be  made  a  little  heavier  than  in  the  four  way 
belt  type,  .6  of  the  percentage  given  for  beams  being  permissible. 

In  slabs  not  reinforced  with  a  supplemental  cantilever  frame, 
but  with  a  depressed  head  instead,  the  minimum  thickness  should 
not  be  less  than  the  minimum  thickness  allowed  for  the  former 
type,  but  the  steel  ratio  may  be  made  .55  that  for  beams  instead  of 
.5  which  is  permissible  with  mushroom  heads. 


PERMISSIBLE    DEFLECTIONS  297 

In  square  slabs  supported  on  beams  or  walls  the  percentage  of 
steel  in  the  strip  occupying  the  middle  third  of  the  panel  each  way 
should  not  exceed  that  for  beams  of  similar  proportions  of  depth  to 
span.  In  case  the  panel  is  a  rectangular  panel,  the  percentage  of 
steel  should  not  exceed  that  for  square  panel  having  a  side  equal  to 
(2aa  +  62)/(a  +  6)  which  we  have  previously  used  in  determining 
the  minimum  thickness  of  span  for  this  form  of  panel. 

With  the  above  interdependent  limitations  as  to  the  minimum 
thickness  and  maximum  steel  ratio  in  mind  the  designer  has  to  deter- 
mine how  far  in  any  given  design  he  shall  deviate  for  any  reason  from 
them.  By  making  TV  smaller  and  the  slab  consequently  thicker  he 
will  be  able  to  reduce  the  deflection  to  such  figures  as  may  be  desired 
or  required.  But  when  such  deviations  are  made  the  proper  rela- 
tionship between  steel  ratio  and  slab  thickness  is  still  to  be  deter- 
mined from  the  diagram  page  86  as  has  just  been  done  for  the 
case  of  minimum  thickness. 

The  erroneous  requirement  is  made  in  some  building  codes  that  the 
maximum  deflection  of  a  slab  under  test  shall  be  a  no  greater  per- 
centage of  length  of  span  than  that  permitted  or  allowed  for  a  deep 
beam.  A  deep  beam  might  be  broken  and  seriously  injured  under 
a  deflection  which  would  do  no  damage  whatever  to  a  long  thin 
slab.  A  certain  degree  of  stiffness,  however,  is  required  under  work- 
ing loads  wherever  there  are  partitions  in  a  building  which  may  be 
damaged  by  undue  deflections.  The  deflection  in  a  thin  slab  should 
not  exceed  l/700th  of  the  span  under  working  load,  and  preferably 
less.  In  an  office  building  or  an  apartment  house,  where  there  are 
partitions  which  might  be  cracked  by  deflection,  the  deflection  should 
be  limited  for  working  loads  to  l/1000th  of  the  span. 

The  inappropriateness  of  any  requirement  limiting  the  deflection 
to  a  given  fraction  of  the  span  will  appear  from  the  following  inves- 
tigation of  the  elementary  relations  between  the  deflections  and 
stresses  in  the  steel  of  reinforced  beams,  since  similar  considerations 
apply  to  slabs.  Using  the  notation  previously  employed  we  obtain 
from  the  well  known  expression  for  the  steel  stress  /s,  the  equation 
W  L  (I—  k)  d  =  nlfw 

in  which  n  =  4,  8,  8  or  12  for  four  different  cases,  namely  simple 
and  restrained  beams  carrying  a  load  W  either  concentrated  at  mid 
span  or  uniformly  distributed. 

Again  the  well  known  expression  for  the  deflection  D  gives  us 
the  equation. 


298  SIZE    AND    SPACING    OF    SLAB    RODS 

W  L3  =  m  E  I  D 
in  which  m  =  48,  76.8,  192  or  384  for  the  four  cases  mentioned. 

By   combining   these   equations   to   eliminate    W   the   following 
relation  between  fs  and  D  is  obtained: 

' 


m  (1  —  k)dE 

This  shows  that  the  maximum  deflection  D  in  beams  of  different 
depths  d,  and  the  same  length  and  unit  steel  stress  /s  varies  as  1/d 
while  the  coefficient  n  and  m  make  further  wide  variations,  which 
make  it  absurd  to  limit  the  permissible  deflection  to  any  given  frac- 
tion of  the  span. 

For  test  loads,  however,  greater  deflections  are  permissible  than 
for  the  working  loads  discussed  previously.  Actual  test  under 
common  conditions  of  partial  restraint  shows  that  in  a  span  of  forty 
times  the  thickness  or  depth  of  the  slab  i.  e.,  L  =  40  d,  a  deflection 
equal  to  L/250  will  not  materially  injure  the  construction.  Con- 
sequently in  case  L  =  10  d,  a  deflection  of  L/1000  should  not  be 
exceeded. 

Since  a  practical  constructor  desires  to  test  only  to  safe  limits 
rather  than  to  anything  approaching  the  ultimate  limit  he  will  not 
object  to  regulations  twenty  percent  more  rigid  than  those  just  men- 
tioned provided  the  time  of  making  the  test  is  not  less  than  four 
months  after  casting  the  concrete. 

7.     Size  and  Spacing  of  Rods  in  Flat  Slab   Construction.     The 

plate  action  which  we  have  been  treating,  brought  about  by  indirect 
stress,  depends  for  its  efficiency  upon  the  dissemination  of  the  steel 
through  the  mass  of  the  concrete.  Large  rods  and  wide  spacing 
should  accordingly  be  avoided.  Where  the  steel  percentage  is  very 
small  it  sometimes  happens  that  5/16  or  3/8  inch  rods,  the  smallest 
practical  sizes,  will  be  spaced  as  far  apart  as  eight  or  nine  inch  centers. 
While  reasonable  results  will  be  secured  with  such  spacing  where 
the  percentage  of  steel  in  the  belt  is  as  small  as  .22,  where  the  per- 
cent of  steel  is  larger  the  spacing  should  be  closer.  Three-eighths  rods 
4"  centers  are  preferable  to  7/16"  rods  at  six  inch  centers  and  far 
better  than  half  inch  rods  at  eight  inch  centers. 

One  of  the  common  errors  in  flat  slab  design  is  the  use  of  such 
rods  as  1/2  inch  or  even  5/8  inch  from  eight  to  twelve  inches  be- 
tween centers  with  the  expectation  of  securing  results  in  keeping 
with  a  more  scientific  and  uniform  dissemination  of  metal. 


HOLLOW    TILE    WITH    TWO-WAY    REINFORCEMENT  299 

8.  Rectangular  Panels  of  Hollow  Tile  and  Concrete  with  Two= 
Way  Reinforcement  and  Supported  on  Side  Beams.  This  com- 
bination has  been  used  primarily  with  the  idea  of  reducing  dead  weight 
and  of  securing  greater  depth  of  slab.  The  construction  consists  in 
reality  of  a  network  of  narrow  concrete  beams  or  ribs  filling  the 
spaces  between  hollow  tile  blocks  which  usually  are  about  12  x  12 
inches  horizontally.  These  beams  should  preferably  be  not  less  than 
five  inches  in  width  and  should  be  reinforced  with  at  least  two  rods 
each,  one  at  the  bottom  thruout  and  lapping  completely  over  the 
supporting  beams  while  the  other  is  bent  up  over  the  top  and  given 
a  lap  of  at  least  a  foot  or  two  beyond  the  beams. 

It  is  customary  to  put  the  same  reinforcement  in  each  of  the  ribs 
regardless  of  its  position  in  the  slab,  thus  giving  a  uniform  reinforce- 
ment thruout  the  slab. 

Such  a  construction  will  be  properly  figured  as  a  beam  construc- 
tion, treating  it  as  continuous  so  far  as  dead  load  is  concerned  pro- 
vided it  has  ample  laps,  and  as  a  simple  beam  as  respects  its  live 
loads. 

In  one  form  of  two-way  tile  construction  the  ends  of  the  hollows 
in  the  blocks  are  closed  by  U-shaped  pieces  of  terra  cotta  thus 
leaving  a  rectangular  net-work  of  crisscross  channels  to  be  filled  in 
with  reinforced  concrete  which  also  spreads  over  the  entire  top 
surface  in  a  layer  several  inches  in  thickness.  This  forms  a  rec- 
tangular net  work  of  T-beams  or  ribs,  but  the  continuity  of  the  lower 
part  of  the  slab  is  so  imperfect  as  to  transmit  no  more  than  a  negligible 
amount  of  indirect  tensile  stress  from  rod  to  rod,  especially  under 
heavy  loads.  What  it  may  do  under  light  loads  is  of  no  account  in 
design.  Tho  some  little  coaction  might  possibly  occur  thru  bond 
shear  at  the  intersections  of  the  ribs,  the  tile  blocks  are  as  a  rule 
twelve  inches  in  width  which  puts  the  reinforcing  rods  some  sixteen 
to  seventeen  inches  apart  so  that  the  structure  with  this  wide  spac- 
ing is  not  sufficiently  fine  grained  or  uniform  in  texture  to  approxi- 
mate in  effect  to  the  properties  or  characteristics  of  a  homogeneous 
plate.  It  seems  conservative  therefore  to  treat  this  combination 
on  the  beam  strip  theory  as  provided  in  most  building  codes. 

In  case  of  a  live  load  W1  uniformly  distributed  on  a  square  panel, 
consider  the  four  triangular  areas  into  which  it  is  divided  by  the 
diagonals.  Two  of  these  may  be  taken  to  be  transmitted  to  the 
sides  by  one  set  of  ribs,  and  two  by  the  other.  Since  the  center 
of  gravity  of  each  triangle  is  at  a  distance  L/6  from  the  edge  and 
its  load  is  Wi/4  the  total  applied  moment  across  the  slab  at  mid 
section  is 


300  DEFLECTION    AND    STEEL    STRESSES    WITH    HOLLOW    TILE 

i  W,  (L/2  —  L/3)  ==  Wi  L/24. 

instead  of  TFjL/16  as  would  occur  in  case  the  load  W\/2  carried 
by  this  parallel  set  of  ribs  were  uniformly  distributed  along  them. 
The  several  parallel  ribs  do  not  resist  this  moment  equally,  but  for 
a  considerable  portion  of  the  width  there  is  little  difference  in  this 
respect.  The  basis  of  computation,  being  that  of  a  simple  beam,  is 
so  liberal  that  the  applied  moment  }\\  L/24  is  ample. 

Again,  in  case  of  a  dead  load  of  W2  on  the  panel  the  building 
codes  of  various  cities  prescribe  that  it  shall  be  computed  on  the 
same  basis  as  the  live  loads.  But  since  the  results  so  obtained  are 
not  thot  to  be  in  good  accordance  with  experiment  it  would  seem 
preferable  to  apply  the  equations  already  derived  for  the  case  of  a 
concrete  slab  supported  on  side  beams  to  this  case.  Stiff  side  beams 
cause  the  formation  of  valleys  in  this  case  as  they  do  in  any  slab  sup- 
ported in  this  manner.  But  with  tile  the  reinforcing  rods  in  the 
two  directions  act  practically  independently  of  each  other  at  heavy 
loads.  It  will  therefore  not  be  possible  to  assume  that  they  in  effect 
form  a  single  sheet  of  steel  as  has  been  done  where  belts  of  rods  cross 
each  other  in  solid  embedment.  The  steel  will  in  this  case  therefore 
be  only  one  half  as  effective.  Moreover  the  value  of  K  will  be  so 
small  that  its  square  is  negligible. 

Make  these  modifications  in  equations  (30)  and  (31)  by  first 
multiplying  them  by  4/3  in  order  to  remove  the  effect  of  the  embed- 
ment which  was  included  in  them  and  then  replace  2A8  by  J  S  As 
because  the  belts  act  independently,  and  we  have 

rr  2  J-^2        1 

D  =  -  -  ...........................  (32) 

.68  X  1010f/2Z  As 

W2  L\/L2 
U  =  ~  ............................  (33) 


For  a  square  it  will  be  noticed  that  on  this  basis  the  unit  stress  per 
pound  of  live  load  is  twice  that  per  pound  of  dead  load  in  case  of 
A1  =  A2  =  JS  As,  because  the  former  is  taken  to  be  carried  by 
simple  beam  action  and  the  latter  by  beams  that  are  mutually  res- 
trained. 

We  may  then  on  analogy  of  (33)  write  the  total  unit  stress  as 
follows  : 


....................  (34) 

12  j  d  2  As 


PROPERTIES    OF    EXPANDED    METAL    REINFORCEMENT  301 

in  which  the  effect  of  the  relative  length  of  the  sides  is  introduced 
for  both  live  and  dead  loads  in  the  same  manner.  It  is  not  advisable 
in  this  construction  to  make  L2/Ll  less  than  0.8. 

The  observed  deflections  under  a  moderate  test  load  of  any  amount 
W2  will  be  less  than  those  computed  from  equation  (32),  and  the 
results  observed  will  not  reach  those  computed  until  the  stress  in  the 
steel  reaches  approximately  eight-tenths  of  its  yield  point  value. 

9.  Expanded  Metal  as  a  Form  of  Reinforcement  for  Slabs. 
Expanded  metal  is  formed  by  slashing  sheets  of  steel  in  such  manner 
that  it  may  be  stretched  or  expanded  laterally  to  form  a  diamond 
mesh.  The  junction  of  one  mesh  to  another  beside  it  is  called  a 
bridge,  and  in  forming  the  bridge  the  metal  is  not  only  distorted  in 
a  lateral  direction  but  twisted  at  the  bridge.  The  diamond  meshes 
thus  formed  are  usually  exactly  or  approximately  twice  the  length 
of  the  short  diameter  of  the  mesh.  This  proportion  gives  a  right 
cross  section  of  metal  half  as  great  laterally  as  longitudinally  and 
in  view  of  the  twisting  the  stiffness  laterally  is  very  small  indeed 
in  comparison  with  the  stiffness  in  a  longitudinal  direction.  When 
embedded  in  concrete  expanded  metal  forms  a  very  convenient  means 
of  reinforcement  for  short  spans  and  light  loads.  It  must  be  placed 
with  the  long  direction  of  the  mesh  in  the  line  of  greatest  stress 
since  if  it  were  placed  in  a  lateral  instead  of  in  a  longitudinal  direc- 
tion not  only  would  the  effective  cross  section  of  the  metal  be  reduced 
one  half  but  its  efficiency  would  be  further  reduced  one  half  because 
of  the  unfavorable  angle  of  inclination  of  the  strand  to  the  direction 
of  the  stress  and  even  this  value  is  further  much  reduced  because 
the  consecutive  diamonds  or  meshes  do  not  pull  in  line  laterally  on 
account  of  the  offset  at  the  bridge.  The  net  effect  of  the  form  of 
the  mesh,  the  reduced  section  of  metal  resisting  strain  in  a  lateral 
direction  and  the  offset  at  the  bridge,  accordingly  is  to  reduce  the 
lateral  efficiency  of  the  sheet  as  a  means  of  reinforcement  to  less  than 
25%  of  its  value  longitudinally  and  probably  to  about  ten  or  twelve 
percent  in  view  of  the  bridge.  All  catalogues  published  by  manu- 
facturers of  expanded  metal  give  explicit  instructions  that  the  long 
dimension  of  the  diamond  shall  be  placed  in  the  direction  of  the 
greater  stress  wherever  this  material  is  used  as  a  means  of  reinforcing 
concrete  slab  construction. 

The  coaction  of  expanded  metal  with  concrete  differs  greatly 
from  that  of  a  diamond  mesh  formed  of  straight  rods  in  two  layers 
crossing  each  other  diagonally  for  the  reason  that  in  the  case  of  the 
rods  shear  resistance  between  the  two  layers  is  furnished  by  bond 
shear  between  the  concrete  and  the  respective  rods,  whereas,  in  the 


.302  EXPANDED    METAL    NOT    EQUIVALENT    TO    RODS 

case  of  expanded  metal  the  bridge  forms  a  rigid  connection  and  this 
shear  is  developed  largely,  or  almost  entirely  in  the  steel  itself. 
Thus  the  coaction  of  expanded  metal  with  concrete  differs  radically 
from  that  of  rod  reinforcement  in  the  manner  in  which  energy  is 
stored  by  the  deformations  of  the  concrete  and  metal.  The  bond 
between  expanded  metal  and  the  concrete  is  formed  by  interlocking 
blocks  formed  within  the  mesh  of  the  metal  and  as  stress  is  brought 
upon  the  metal  in  the  direction  of  the  length  of  the  sheet  these  blocks 
are  compressed  laterally  and  the  tendency  of  one  mesh  to  slide  upon 
another  is  prevented  by  the  rigid  steel  section  of  the  bridge. 

Efforts  were  made  early  in  1901  to  construct  flat  slab  floors 
reinforced  with  expanded  metal.  Sheets  of  expanded  metal  were 
laid  nearly  covering  the  bottom  of  the  slab  from  column  to  column 
and  sheets  were  placed  at  the  top  of  the  slab  over  the  columns  in 
the  form  of  a  cross,  and  it  is  claimed  that  sheets  were  also  placed 
in  parallel  position  over  column  tops.  Arranged  in  the  form  of  a 
cross  the  sheets  were  not  effective  as  a  circumferential  cantilever 
reinforcement.  Since  the  ends  of  the  sheets  forming  the  cross  were 
arranged  to  project  far  beyond  the  over  lapping  area  it  was  man- 
ifestly impossible  to  approximate  plate  action  in  this  manner  so  that 
the  few  structures  so  erected  developed  no  greater  carrying  capacity 
than  would  be  expected  on  beam  strip  theory. 

Expanded  metal  in  coaction  with  the  concrete,  adds,  however, 
a  greater  increase  in  strength  than  the  mere  cross  section  and  yield 
point  value  of  the  metal  would  indicate.  This  would  appear  to  be 
due  to  the  storing  up  of  energy  in  the  slab  by  the  lateral  compression 
of  the  concrete  thru  longitudinal  strain  in  the  mesh. 

Wire  woven  fabrics  also  are  often  a  most  convenient  form  of 
reinforcement  and  frequently  the  best  possible  to  be  used  for  wrapping 
as  reinforcement  for  beams,  light  slabs,  roof  work  and  the  like,  and 
altho  their  cost  per  pound  is  greater  than  rods  convenience  in  hand- 
ling and  placing  may  more  than  offset  this  difference.  The  fact 
that  thoro  dissemination  of  metal  in  small  units  thru  the  concrete 
is  conducive  to  better  results  than  the  use  of  the  same  sectional  area 
in  larger  units  makes  their  use  for  many  purposes  almost  indis- 
pensible. 


303 


CHAPTER  VIII 
REINFORCED  CONCRETE  COLUMNS 

1.  General  Considerations.  The  requirements  for  suitable 
design  for  reinforced  concrete  columns  in  building  construction 
may  be  briefly  stated  as  follows: 

First,  that  the  longitudinal  reinforcing  metal  should  be  toward 
the  outer  portion  of  the  column  in  order  to  properly  resist  any 
tendency  to  bend  or  deflect. 

Second,  that  the  bars  should  be  banded  or  tied  together  to  main- 
tain them  in  their  desired  position  and  add  toughness  to  the  column. 

Third,  that  the  bands  or  ties  should  not  so  cross  the  core  as  to 
interfere  with  placing  the  concrete  and  securing  a  monolithic  solid 
core. 

Fig.  Type  A,  shows  one  of  the  old  forms  of  Hennebique  type  of 
column.  In  this  type  the  principal  reinforcement  consists  of  heavy 
vertical  or  longitudinal  bars  tied  across  laterally  from  one  to  the 
other  by  comparatively  small  ties.  It  will  be  noted  that  these  ties 
cross  and  recross  the  core  of  the  column,  and  require  considerable 
care  in  filling  to  make  sure  that  there  are  no  voids  in  the  finished 
work. 


Hr~ 

I     'l 

- 


Fijr.  Type  A   Column 


Fig.   Type  B  Column 


304 


TYPES   OF    COLUMXS 


Cases  have  occurred  with  this  type  where  the  concrete  has  been 
arrested  part  way  down  in  pouring  the  column  and  on  removal  of 
the  form  an  open  space  was  found  of  perhaps  as  much  as  two  feet 
between  the  concrete  above  and  below,  so  that  the  load  above  was 
carried  by  the  vertical  bars  only.  Evidently  such  an  arrange- 
ment of  metal  is  somewhat  dangerous,  but  with  unusual  care  it  may 
prove  satisfactory  from  the  standpoint  of  strength  if  the  work  be 
properly  executed. 

Fig.  Type  B,  shows  an  improved  form  of  tying  together  the 
eight  vertical  bars  forming  the  vertical  reinforcement  with  horizontal 
ties  in  the  form  of  squares,  one  inscribed  within  the  other.  The 
advantage  of  this  type  over  that  previously  shown  lies  in  the  fact 
that  the  central  core  of  the  column,  or  inscribed  square,  is  clear  and 
unobstructed  thruout . 

Fig.  Type  C,  shows  a  column  reinforcement  consisting  of  four 
vertical  rods  with  wrapping  or  ties  holding  them  together  at  inter- 
vals. This  is  suitable  for  very  light  loads  where  the  concrete  is 
more  than  sufficient  to  take  the  entire  compression  without  excessive 
stress. 

Fig.  Type  D,  shows  a  column  section  of  the  Considere  type  in 
which  the  vertical  rods  are  hooped  with  spiral  reinforcement.  Con- 
siderable work  has  been  executed  using  hooped  columns  that  omit 
the  vertical  steel.  This,  as  the  authors  view  it,  is  a  very  grave 
mistake. 


I 


Fig.  Type   D  Column  Considere  Type.     Fig.  Type  C   Column  Suitable  for  Light  Loads  Only. 


TYPES    OF    COLUMNS  305 

Hooping  may  be  of  two  types:  First,  a  spiral  coil  in  which 
the  wire  is  wound  around  the  core  of  the  column  in  the  form  of  a 
continuous  spiral,  and  second  that  in  which  separate  independent 
hoops  are  placed  at  intervals  and  attached  to  the  vertical  rein- 
forcement. 

The  strength  of  Types  A,  B,  C,  and  D,  all  depend,  first  upon 
the  strength  of  the  concrete,  second,  upon  the  amount  of  vertical 
steel  used,  and,  third,  upon  the  amount  of  ties  or  hooping  holding 
the  rods  in  position  and  bringing  lateral  restraint  upon  the  concrete. 

Theoretical  formula  based  on  the  ratio  of  the  moduli  of  elas- 
ticity of  concrete  and  steel  alone  cannot  be  depended  upon  for  a 
satisfactory  solution  of  the  problem  presented  by  the  third  element 
noted  and  we  must  depend  largely  upon  experimental  investigation 
to  determine  reasonable  and  safe  practical  values  to  use  for  our 
working  stress. 

In  deciding  upon  these  values  we  need  to  consider  the  column, 
first,  from  the  standpoint  of  its  ultimate  strength  in  the  finished 
building,  second,  from  the  standpoint  of  its  strength  and  safety 
during  construction,  and  third,  from  a  consideration  of  the  relative 
values  of  the  various  types  in  securing  strength  at  a  minimum  cost. 

Type  D,  with  a  proper  proportion  of  vertical  steel  combined 
with  the  hooping  ranks  first,  from  the  standpoint  of  safety  and 
economy. 

Type  B,  second. 

Type  D,  with  hooping  but  with  no  vertical  steel  third;  and  types 
A  and  C  fourth. 

It  may  be  stated  that  type  A  is  now  rarely  used  and  discussion 
concerning  it  may  be  omitted. 

For  Type  C,  the  allowance  permissible  for  working  stress  on  a 
1:2:4  concrete  is  350  pounds  per  inch  of  the  core  area  between  rods, 
and  10,000  pounds  per  square  inch  on  the  vertical  steel  and  besides 
this  the  volume  of  metal  in  the  ties  is  to  be  treated  as  forming  im- 
aginary verticals  with  a  working  stress  equal  to  that  allowed  for  the 
vertical  bars,  the  ties  to  be  spaced  not  further  apart  than  ten  times 
the  diameter  of  the  vertical  bars  in  case  the  bars  are  one  inch  sec- 
tion, but  where  smaller  bars  are  used  the  spacing  should  not  exceed 
9"  nor  the  size  of  the  tie  to  be  less  than  one  quarter  inch  round. 

Type  B.  The  allowable  working  stress  for  a  1:2:4  mix  is 
600  pounds  per  square  inch  on  the  concrete  of  the  core,  10,000  pounds 
on  the  vertical  steel,  and  one  and  one  half  times  the  volume  of  the 
ties  treated  as  imaginary  verticals.  These  ties  should  not  be  spread 


306  CONSIDERE    TYPE 

further  apart  than  9",  and,  if  they  are  to  be  considered  of  value, 
they  should  be  put  not  more  than  ten  diameters  of  the  vertical  bars 
apart. 

Type  D,  the  Considere  type,  is  by  far  the  most  economical  type 
of  column  reinforcement  that  has  been  invented.  It  was  brought 
prominently  to  the  attention  of  the  public  by  Armand  Considere. 

The  principle  involved  is  this:  by  restraining  the  concrete 
laterally  its  strength  in  compression  is  greatly  increased.  Just  as 
an  ordinary  piece  of  stove  pipe  filled  with  sand  will  carry  a  load 
several  times  greater  than  the  pipe  itself  would  be  able  to  do;  so  will 
a  hooped  column  owing  to  the  fact  that  the  metal  is  strained  in 
tension,  while  the  rilling  held  in  position  by  the  restraint  of  the  pipe, 
carries  the  weight  of  the  load.  For  strength  see  Section  5. 

There  have  been  quite  a  number  of  experiments  on  hooped  con- 
crete using  spiral  hooping  only.  In  these  experiments  it  has  been 
found  that  after  the  ultimate  strength  of  plain  concrete  has  been 
developed,  splitting  and  scaling  of  the  outside  shell  occurs,  combined 
with  a  large  vertical  deformation  and  considerable  lateral  bending 
before  ultimate  failure. 

2.  Considerations  of  Safety  Determining  Carrying  Loads. 
If  it  is  expected  to  develop  the  core  of  the  concrete  to  a  point  beyond 
its  normal  strength  we  must  evidently  prevent  its  lateral  distortion 
or  bulging  and  also  the  sliding  or  flow  of  the  concrete  between 
consecutive  bands  or  turns  of  the  spiral,  hence  a  certain  proportion 
of  vertical  steel  must  be  used  in  connection  with  the  hooping  to  secure 
the  best  results. 

In  determining  the  degree  of  safety  of  the  various  types  of  column 
design,  an  investigation  of  the  manner  of  failure  of  the  respective 
types  is  in  order  as  to  whether  it  occurs  suddenly  and  without 
warning,  or  gradually,  accompanied  by  indications  of  approaching 
failure  long  before  failure  occurs,  and  there  is  the  further  question 
as  to  whether  the  conditions  of  strain  in  the  column  are  proportional 
or  comparable  between  the  column  under  ordinary  working  stresses 
and  the  column  as  it  approaches  the  breaking  down  point  and 
ultimate  strength.  The  following  general  observations  may  be 
made  as  to  these,  questions: 

In  columns  shown  in  Fig.  Type  C,  the  failure  occurs  with  little 
warning,  the  vertical  bars  bending  outward  and  the  ties  yielding. 

In  the  hooped  column  without  vertical  steel,  when  it  is  loaded 
from  forty  to  fifty  percent  of  its  ultimate  strength,  the  portion  of 
the  concrete  outside  the  hooping  commences  to  check  and  crack, 


TESTS    OF    HOOPED    AND    VERTICALLY    REINFORCED    COLUMNS  307 

and  later  to  scale.  From  this  point,  the  rate  of  deformation  with 
addition  of  the  load  increases  rapidly  owing  to  dissipation  of  energy 
by  the  cracking  and  scaling  of  the  shell.  Further  loading  is  ac- 
companied by  large  lateral  deformations  up  to  the  final  failure.  Such 
a  column  gives  ample  warning  but  the  point  at  which  the  outer  shell 
or  fire  protection  commences  to  fail  is  but  little  higher  than  the 
point  at  which  the  ordinary  vertically  reinforced  column  fails,  so 
that  little  advantage  in  the  way  of  increased  working  stress  is 
secured  unless  the  hooping  is  combined  with  vertical  steel. 

The  well  hooped  column  vertically  reinforced  shows  a  large 
increase  in  strength  over  that  of  the  vertically  reinforced  column 
with  ties  and  a  great  increase  in  toughness.  Its  failure  is  not  sudden 
and  without  warning  as  in  the  former  type,  while  the  point  at  which 
checking  and  scaling  of  the  outside  shell  occurs  is  raised  to  eighty 
or  eighty-five  percent  of  the  ultimate  strength,  thus  giving  a  large 
margin  of  safety  to  the  fireproofing  between  the  working  load  and 
the  load  where  the  failure  of  the  shell  is  in  evidence. 

3.  Experimental  Data.  A  partial  report  on  tests  on  full  sized 
columns,  made  at  Phoenixville,  Pa.,  for  C.  A.  P.  Turner,  engineer, 
by  Mason  D.  Pratt,  is  given  in  the  following  table: 

Test  No.  1 

Marks  on  column:     None. 

Reinforcement:     Eight  1J  inch  round  vertical  bars. 

Bands:     Spaced  9  inches  vertically,  |  inch  rivets,  cross  section 
If  x  }  inches,  inside  diameter  14  inches. 

Hooped  with  7/32  inch  wire  spirals,  about  2  inch  pitch. 

Total  load  at  failure,  1,360,000  Ibs. 

Remarks:  Point  of  failure  was  about  22  inches  from  the  top. 
Little  indication  of  failure  until  ultimate  load  was  reached. 

Some  slight  breaking  off  of  concrete  near  the  top  cap,  due 
possibly  to  the  cap  not  being  well  seated  in  the  column  itself. 

Test  No.  2 

Marks  on  columns:     Box  4. 

Reinforcement:     Eight  If  inch  round  vertical  bars. 

Bands:     Spaced  13  inches  vertically,  \  inch  rivets,  cross  section 
If  x  J  inches,  inside  diameter  14  inches. 

Hooped  with  7/32  inch  wire  spiral,  about  3  inch  pitch. 

Point  of  failure:     About  18  inches  from  top. 

Top  of  cast  iron  cap  cracked  at  four  corners. 

Ultimate  load:     1,260,000  Ibs. 

Remarks:  Both  caps  apparently  well  seated,  as  was  the  case 
with  all  the  subsequent  tests. 


308 


COLUMN    TESTS    AT    PHOENIXVILLE 


Test  No.    1 
Showing  Column  as  it  Came  from  Testing  Machine. 


Test  No.  3 

Marks  on  column:     4-b. 
Reinforcement:     Eight  7/8  inch  round  vertical  bars. 

Bands:     Spaced  13  inches  vertically,  J  inch  rivets,  cross  section 
If  x  3/16  inches,  outside  diameter  14  inches. 

Ultimate  load:     900,000  Ibs. 

Point  of  failure:     About  2  feet  from  top. 

Remarks:     Concrete  at  failure,  considerably  disintegrated,  prob- 
ably due  to  continuance  of  movement  of  machine  after  failure. 


COLUMN    TESTS    AT    PHOENIXVILLE 


309 


Test  No.  4 

Marks  on  columns:     Box  4-c. 

Reinforcement:     Eight  1  inch  round  vertical  bars. 

Bands:  Spaced  8  inches  vertically,  J  inch  rivets,  cross  section 
If  x  J  inches,  inside  diameter  14  inches. 

Hooped  with  7/32  inch  wire  spirals,  about  3  inch  pitch. 

Total  load  at  failure:     1,260,000  Ibs. 

Remarks:  First  indications  of  failure  were  nearest  the  bottom 
end  of  the  column,  but  the  total  failure  was  as  in  all  columns,  within 
2  feet  of  the  top.  Large  cracks  in  the  shell  of  the  column  extended 
from  both  ends  to  very  near  the  middle.  This  was  the  most  satis- 
factory showing  of  all  the  columns,  as  the  failure  extended  over 
nearly  the  full  length  of  the  column. 


Column   No.  4,  after  Tost 

Test  No.  5 

Marks  on  column:     None. 
Reinforcement:     Eight  |  inch  vertical  bars. 
Bands:     Spaced  10  inches  vertically,  J  inch  rivets,  cross  section 

If  x  J  inches,  14 \  inches  outside  diameter. 
Hooped  with  7/32  inch  wire  spiral  as  before,  3  inch  pitch. 


310  CHARACTER    OF    FAILURE 

Load  at  failure:  1,100,000  Ibs. 
Ultimate  load:     1,130,000  Ibs. 

Remarks:  The  main  point  of  failure  in  this  as  in  all  other  col- 
umns was  within  two  feet  of  the  top  altho  this  column  showed  some 
scaling  off  at  the  lower  end. 

This  set  of  tests  were  not  conducted  with  any  considerable 
degree  of  refinement  but  were  a  practical  test  of  ultimate  strength 
and  the  yield  point  value  of  specimens  of  full  sized  members, 
which  lends  greater  value  to  the  determination  than  laboratory 
tests  on  small  specimens. 

The  concrete  mixture  was  one  part  Portland  cement,  one  part 
sand,  and  one  and  one-half  parts  buckwheat  gravel,  and  three  and 
one-half  parts  gravel  ranging  from  one-quarter  inch  to  three- 
quarter  inch  in  size. 

It  should  be  noted  that  in  these  tests  the  cracking  of  the  shell 
did  not  occur  until  the  hoops  were  over-strained,  and  that  the 
strength  of  the  hooping  closely  defined  the  ultimate  strength  of  the 
column  with  the  proportions  of  vertical  steel  used. 

1.  In  these  columns,  pressures  were  developed  on  the  core  more 
than  three  times  the  ultimate  strength  of  plain  concrete  at  2600 
pounds  per  square  inch. 

2.  Incipient  failure  occurred  only  by  the  stretching  or  bursting 
of  the  bands. 

All  columns  were  approximately  octagonal  in  shape,  10'6"  long 
and  18"  diameter.  Final  failure  occurred  toward  the  upper  end  of 
the  column.  Mr.  Pratt  accounts  for  the  regularity  with  which  the 
columns  failed  at  the  upper  end  on  the  ground  that  the  concrete 
at  the  lower  end  was  more  dense,  owing  to  its  being  under  consider- 
able hydraulic  pressure,  while  setting.  The  rods  were  all  shorter 
than  the  concrete  shaft.  Examination  of  the  columns  after  removal 
from  the  testing  machine  showed  in  all  cases  a  bulging  out  of  the 
vertical  reinforcement  at  the  principal  point  of  failure  with  the 
nearest  hoop  ruptured  and  in  every  case  the  wire  spiral  was  broken 
in  one  or  more  coils  at  the  point  where  the  vertical  rods  were  bent 
out.  The  vertical  bars  in  nearly  every  case  bulged  as  a  column 
with  fixed  ends.  Where  the  hoop  spacing  was  six  to  nine  inches, 
the  deformed  length  of  the  bar  would  extend  over  the  space  of 
two  hoops.  Where  the  hoop  spacing  exceeded  nine  inches  the 
deformed  length  of  vertical  bars  was  confined  to  the  space  between 
one  pair  of  hoops. 


BACH  S  TESTS TYPE  C  COLUMNS 


311 


It  may  be  well  to  review  at  this  point  two  series  of  carefully 
conducted  tests  upon  plain  and  reinforced  concrete  prisms.  Fig.  76, 
shows  the  dimensions  and  reinforcement  of  a  series  of  prisms  tested 
by  Bach*  for  Wayss  and  Freitag.  These  specimens  are  Type  C 
columns. 


n'*'l      n 

r 

V 

1 

V 

5 

k^ 

t) 

^ 

s 

II 

1 

1 

1 

i 

LJ 

v  I 

_ 

T 

i 

cJ& 

-rl 

PPSI 

i$m 

^ 
K 

I 

1 

?^kMoMtf 

-1  j 

-    *&"  ^ 

•I 

<  &  4Q 

Fig.  76  Showing  Dimensions  and  Reinforcement. 

The  first  table  for  elasticity  test  of  Type  C  columns,  shows  the 
reinforcement,  tie  spacing,  and  elastic  deportment  under  different 
loads. 

The  second  table  for  Type  C  columns,  shows  the  breaking 
strength  of  this  series. 

*See  Het  Cement-Ijzer,  Saunders.  p.  91. 
Eisenbetonbau,  Morsch. 


312 


BACH'S  COLUMN  TESTS 


It  will  be  noted  under  the  table  showing  the  breaking  strength 
that  the  specimens  with  1  3/16  inch  round  verticals  and  4.6  per- 
cent reinforcement,  do  not  show  as  great  an  increase  in  strength 
as  the  specimens  with  1.14  percent  reinforcement,  but  with  much 
closer  spacing  of  the  ties,  the  specimens  with  the  large  rods  having 
ties  four  times  as  far  apart  as  the  specimens  with  the  small  rods. 

COL.  TABLE  I 

Elasticity  Test  of  Columns  (Bach) 


4  Rods  of               Tie 

Shortening  in  Millionths 

Stresses 

Diameter          Spacing 

of  the  length 

Lbs.  sq.  in. 

Inch               Inches 

Total 

Elastic  Diff. 

Permanent  Set 

459 

Plain  Concrete 

133 

7 

126 

459 

1  in.    rd.         10 

114 

5 

109 

459 

I  "       "             5 

110 

2 

108 

459 

1   "       "             2| 

106 

4 

102 

919 

Plain  Concrete 

333 

37 

296 

919 

?  in.    rd.          10 

267 

20 

247 

919 

J  «       «              5 

264 

18 

246 

919 

1   "       "             2^ 

241 

13 

228 

1380 

Plain  Concrete 

709 

164 

545 

1380 

-!|  in.    rd.          10 

488 

63 

425 

1380 

I   "        "              5 

473 

58 

415 

1380 

5     a          tt                  91 

421 

42 

379 

COL.  TABLE  II. 

Breaking  Strength  of  Columns  (Bach) 


Specimen  3  mo.  old. 

Breaking  Strength 

Diameter 
of  rods. 

Tie 
Spacing 

Each 

Average 
Ibs.  per  sq.  in. 

Percent   of 
Reinforcing 

Plain 

Concrete 

2076 

1963 

1977 

2006 

0 

-|  in.  rd. 

10    in. 

2432 

2290 

2447 

2390 

1.14 

1  "       " 

5     " 

2389 

2660 

2489 

2512 

1.14 

1   "       " 

2-  " 

3015 

2845 

2887 

2915 

1.14 

3      «              « 
4 

10     " 

2404 

2404 

2446 

2418 

2.04 

1&"              " 

10     " 

2474 

2830 

2802 

2702 

4.60 

Test  Cubes  . 

2389 

2404 
2631 

2432 
2617 

2490 

0 

Looking  at  the  elasticity  tests  of  the  specimen  reinforced  with 
the  same  vertical  steel  the  marked  difference  in  toughness,  increase 
of  modulus  of  elasticity  and  the  reduction  of  permanent  set  pro- 
duced by  the  increase  in  the  number  of  ties,  is  strikingly  shown  by 
this  series  of  tests. 


DANGER    OF    COMMON    FORMULA  313 

This  series  of  tests  show  conclusively  that  where  the  vertical  bars 
do  not  bear  directly  upon  the  face  plate  of  the  machine  and  the  load 
is  brought  upon  the  reinforcement  under  the  usual  condition  of  the 
column  in  the  reinforced  concrete  structure,  that  is  the  steel  is 
strained  by  the  load  brought  on  it  thru  the  concrete,  that  a  formula 
which  does  not  take  into  consideration  the  amount  and  spacing  of 
the  ties  fails  to  account  for  the  deportment  of  the  column  both  as 
regards  elasticity  and  ultimate  resistance.  That  is,  merely  an 
increase  in  the  total  cross  section  of  the  longitudinal  reinforcement 
does  not  produce  an  increase  in  the  breaking  strength  to  the  extent 
which  would  be  expected  by  the  formula 
P  =  /„  (A,  +  nA,) 

Hence  in  inexperienced  hands,  this  formula  may  produce  unsafe 
designs  by  increasing  the  percentage  of  longitudinal  reinforcement 
disproportionately  in  order  to  secure  columns  of  small  diameter. 
This  procedure  gives  a  column  with  a  calculated  margin  of  safety 
which  it  does  not  possess. 

When  the  increase  in  the  resistance  is  compared  for  equal  longi- 
tudinal reinforcement,  there  being  a  difference  in  the  number  of 
ties,  it  is  shown  by  these  experiments  that  the  steel  used  as  ties  is 
much  more  effective  than  the  longitudinal  steel.  As  noted,  however, 
the  columns  reinforced  with  vertical  steel  and  ties  do  not  develop 
that  degree  of  strength  which  is  secured  by  columns  properly  rein- 
forced vertically  and  hooped  with  bands  or  spirals.  Hence  their 
use  has  been  largely  discontinued  where  loads  are  at  all  heavy  and 
is  confined  to  those  cases  where  loads  are  relatively  light  and  some 
slight  saving  may  be  made  in  the  use  of  ties  over  the  cost  of  spiral 
hooping. 

4.  Reinforced  Columns  Classified  by  the  Manner  in  Which 
Loads  are  Applied .  We  have  called  attention  to  the  type  of  prisms 
which  were  tested  by  Bach  in  which  the  steel  was  cut  short  at  the 
end  of  the  concrete  prism  so  that  the  load  applied  to  the  prism  was 
brought  upon  the  longitudinal  steel  thru  the  bond  shear  between  the 
steel  and  the  concrete.  This  is  the  common  mode  of  arrangement 
of  the  steel  in  the  reinforced  concrete  column  in  the  practical  build- 
ing. Sometimes,  however,  in  order  to  secure  relatively  small  columns 
for  heavy  loads  very  high  percentages  of  vertical  steel  are  used  and 
the  load  is  brought  largely  upon  the  steel  by  direct  bearing  of  steel 
upon  steel.  Evidently  the  coaction  of  the  elements  of  the  composite 
structure  differ  widely  under  these  radically  different  conditions  and 
correspondingly  different  formulas  must  be  applied  to  these  different 
conditions. 


314 


BACH'S  COLUMK  TESTS 


We  have  cited  the  interesting  series  of  tests  by  Bach  on  rec- 
tangular prisms,  plain  and  reinforced  with  longitudinal  steel  and 
ties  and  we  may  profitably  consider  the  corresponding  series  of 
tests  on  hooped  and  longitudinally  reinforced  prisms. 

Fig.  77  gives  the  dimensions  and  type  of  reinforcement  used 
while  table  III  shows  the  results  of  the  tests. 


Fig.  77  Dimensions  of  Columns  and  Type  of  Reinforcement. 


As  in  the  first  series  the  longitudinal  steel  was  cut  short  of  the 
ends  of  the  prisms  so  as  to  secure  normal  coaction  between  the  steel 
and  the  concrete.  Percentage  of  hooping  varied  thru  a  wide  range 
and  the  pitch  of  the  spiral  hooping  was  also  varied  between  wide 
limits.  Unfortunately  the  series  is  lacking  in  a  corresponding  varia- 
tion in  the  percentage  of  vertical  steel. 

On  the  whole,  these  specimens  correspond  approximately  with 
results  indicated  by  the  Considere  formula  and  exceed  these  results 
for  those  specimens  which  approach  desirable  proportions  in  point 
of  spacing  of  the  spiral  and  relation  of  the  vertical  steel  to  the  hoop- 
ing. 


BACH'S  COLUMN  TESTS 


315 


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COMPARISON    OF    TEST    DATA 


Specimens  XIII  and  XIV,  in  which  the  pitch  of  the  spiral  was 
exaggerated  gave  results  less  than  the  formula,  pointing  to  the 
necessity  for  closer  spacing. 

In  specimens  II,  III,  IV,  VIII,  IX,  X,  XI  and  XII,  the  sec- 
tional area  of  the  longitudinal  rods  was  small  and  the  results  were 
consequently  indifferent,  but  the  greater  the  total  weight  of  spiral 
reinforcement  the  higher  were  the  values. 

The  tests  show  that  when  the  spirals  are  increased  in  strength, 
their  pitch  must  be  decreased  and  the  cross  section  and  number 
of  vertical  rods  must  be  increased,  for  with  the  increase  of  spirals  the 
concrete  is  in  a  condition  to  resist  heavier  pressure  and  its  tendency 
to  force  its  way  out  between  the  longitudinal  rods  and  the  spiral 
bands  increase. 

Examining  the  table  of  the  values  obtained,  the  increase  in 
strength  and  yield  point  value  per  one  percent  of  total  steel  i  s  found 
to  be  greatest  in  column  specimen  V,  in  which  the  spiral  reinforce- 
ment is  .88  percent  and  the  vertical  reinforcement  was  1.75.  In 
columns  XII  and  XIII,  with  the  spiral  reinforcement  6.82  percent 
and  the  vertical  reinforcement  1.56  percent,  the  increase  percent  of 
total  reinforcement  at  the  yield  point  was  only  87  pounds  and  the 
increase  of  the  ultimate  strength  per  percent  of  the  ultimate  rein- 
forcement 286  pounds  per  square  inch,  showing  an  improper  propor- 
tion of  the  spiral  and  vertical  reinforcement. 

On  noting  the  pitch  of  the  spiral  it  is  quite  noticeable  that  the 
specimen  with  a  small  pitch  gives  the  highest  resistance  per  percent 
of  steel.  In  none  of  these  specimens,  however,  are  the  longitudinal 
rods  of  sufficient  size  to  secure  the  best  results.  Had  5/8,  3/4  and 
1  inch  rods  been  used  in  place  of  1/4  inch,  7/16  and  1/2  inch, 
much  higher  values  would  have  resulted  from  the  stiffness  of  the 
verticals  which  must  act  as  beams  from  coil  to  coil  in  resisting  out- 
ward or  bulging  pressures. 

The  following  table  gives  the  percentage  of  vertical  steel  and 

hooping  in  the  Phoenixville  tests,  see  Section  3. 

COL.  TABLE  IV 
Phoenixville  Column  Tests.     Specimens  3  mo.  old 


No. 
of 
Test, 

% 
Ver- 
ticals 

% 
Spi- 
ral 

% 
Hoop- 
ing 

% 
Spiral 
and 
Hoop- 
ing 

%        Ultimate 
Total       Tested 
Strength 

Ultimate 
Strength 
Per  Sq. 
In. 

Com- 
puted 
Increase 
per  % 
of  Steel 

Strength 
by 
Con- 
sidere 
Formula 

i! 

2 
3 
4 
5 

5.17 
4.82 
3.12 
4.08 
3.12 

1.54 
1.35 
0 
1.36 
1.36 

.99 
.66 

.72 
1.12 
.90 

2.53 
2.01 

.72 
2.48 
2.26 

7.70 
6.83 
3.84 
6.56 
5.38 

1,360,000 
1,260,000 
900,000 
1,260,000 
1,130,000 

8,850 
7,630 
,  5,800 
8,200 
7,350 

785 
710 
792 
820 

842 

1,291,000 
1,279,000 
898,300 
1,216,000 
1,127,000 

THE  CONSIDERK  FORMULA  FOR  COLUMN'S  317 

From  these  tests  it  may  be  observed  that  the  best  results  are  to 
be  secured  when  the  value  of  the  vertical  steel  as  an  element  of 
strength  is  approximately  the  same  as  the  hooping  according  to 
Considered  formula,  Section  5.  In  other  words,  the  reinforcement 
should  be  so  proportioned  that  the  volume  of  the  hooping  for  this 
type  of  column  should  be  between  the  limits  of  35  and  45  percent  of 
the  volume  of  the  vertical  steel  to  secure  the  best  results  in  increas- 
ing the  yield  point  and  ultimate  strength  of  the  column.  Between 
these  limits  the  yield  point  value  of  the  columns  should  vary  from 
75  to  85  percent  of  the  ultimate  strength  of  the  specimen,  so  that 
due  warning  of  approaching  failure  is  given  by  the  member  while 
ample  margin  exists  between  the  safe  working  stress  and  that  point 
at  which  the  fireproofing  will  commence  to  scale  and  chip. 

5.  Considere  Formula.  As  we  have  pointed  out,  the  Con- 
sidere formula  is  conservatively  applicable  to  the  vertically  rein- 
forced and  hooped  columns  provided  that  the  hooping  and  vertical 
steel  are  properly  proportioned.  With  no  vertical  steel,  the  yield 
point  value  of  the  column  is  not  sufficiently  raised  to  warrant  very 
material  increase  in  safe  working  stress  over  that  of  ordinary  types 
notwithstanding  the  fact  that  there  is  an  increase  in  the  ultimate 
resistance  and  large  deformations  occur  before  ultimate  failure. 

The  Considere  formula  for  column  resistance  is  as  follows: 

P  =--  1.5A0/0+  (JSAS  +  2Af»A'a) 

in  which  Ac  is  the  area  of  the  core,  As  the  area  of  the  longitudinal 
steel,  Jls  is  the  area  obtained  by  dividing  the  volume  of  the  hooping 
by  the  length  of  the  column.  The  coefficient  1.5  is  a  coefficient  which 
Considere  considers  represents  the  effect  of  the  hooping  in  increas- 
ing the  strength  of  the  core,  and  that  this  coefficient  is  a  maximum 
at  1.5  and  that  it  is  less  than  this  value  for  a  percentage  of  hooping 
which  does  not  furnish  a  resistance  equivalent  to  700  pounds  per 
square  inch  lateral  pressure  on  the  column. 

Morsch,  however,  and  also  Saunders,  in  discussing  the  Bach 
tests  where  the  percentage  of  steel  was  low,  seem  to  treat  this  co- 
efficient of  Considere  as  the  ratio  of  the  core  to  the  total  area  of 
the  column  (fireproofing  and  all).  In  most  building  ordinances  this 
coefficient,  however,  is  taken  as  unity.  2.4  times  the  volume  of  the 
hooping  is  figured  as  the  effect  of  the  stress  on  the  hooping  in  increas- 
ing the  crushing  resistance  of  the  core  by  the  lateral  pressure  brought 
about  by  the  stress  in  the  hooping  and  represents  the  resistance 
which  would  be  developed  were  the  hooping  in  the  form  of  cylinders 
and  filled  with  a  granular  mass  such  as  sand  subjected  to  pressure. 
Hence  2AA'S  equals  the  cross  section  of  the  equivalent  imaginary 
verticals. 


318  CONSIDERED    TESTS 

Le  Genie  Civil,  Feb.  9,  and  16, 1907,  reported  a  very  extensive  series 
of  tests  by  Considere  on  260  columns  with  lengths  varying  from 
3.15  in.  to  13  ft.  1J  in.  and  diameters  from  1.8  in.  to  27.5  in.  The 
percentage  of  reinforcement  varied  from  1  to  14,  various  methods  of 
spiral  reinforcement,  including  concentric  spirals  were  tried.  The 
effects  of  richness  of  mixture,  age,  percentage  of  water,  ramming  and 
irregularities  in  workmanship  were  also  studied.  One  specimen 
having  12.9  percent  spiral  and  1.2  percent  longitudinal  reinforce- 
ment, and  made  from  a  mixture  containing  1830  Ib.  of  cement  per 
cu.  yd.  of  sand  sustained  25,600  lb./in.2  at  rupture.  A  similar 
specimen  having  only  6.6  percent  of  spiral  reinforcement  failed  at 
17,200  lb./in.2  at  rupture,  with  a  deformation  of  12  percent.  From 
these  and  previous  tests  Considere  deduced  the  conclusions  that 
the  rupture  load  of  a  spirally  reinfored  concrete  column  exceeds  the 
sum  of  the  three  following  factors: 

1.  The  strength  of  the  plain  concrete  times  1.5. 

2.  The  strength  of  the  longitudinal  rods  stressed  to  their  yield 
point. 

3.  The  strength  of  a  longitudinal  reinforcement  of  2.4  times 
the  weight  of  the  spiral  stressed  to  its  yield  point. 

Turner,  in  the  design  and  guarantee  of  strength  for  working  several 
hundred  thousand  columns  has  used  the  Considere  formula  slightly 
modified, allowing  800  pounds  per  square  inch  on  a  1 :  2:  4  mix,  12,000 
pounds  per  square  inch  on  the  vertical  steel,  16,000  pounds  per  square 
inch  on  the  hooping,  treating  the  spirals  as  imaginary  verticals  having 
a  volume  2.4  times  the  volume  of  the  hooping.  For  these  values 
the  ratio  of  the  length  of  the  column  to  its  diameter  should  not  exceed 
12.  For  a  longer  column  increased  vertical  steel  should  be  used  to 
resist  flexure.  Where  the  vertical  resistance  of  the  concrete  developed 
by  the  hooping  exceeds  2000  pounds  per  square  inch  the  proportion 
of  the  mix  should  be  increased  from  1  :  2  :  4  to  1  :  H  :  3,  and  extra 
care  should  be  used  in  the  selection  of  the  stone  aggregate  to  see  that 
it  is  hard  and  satisfactory.  Screened  gravel  is  preferable  where  high 
working  pressures  are  used  in  the  columns.  To  be  efficient,  spiral 
hooping  should  not  exceed  a  4"  pitch  when  combined  with  not  less 
than  four  vertical  rods  and  these  should  be  limited  to  a  minimum 
diameter  of  3/4  of  an  inch  for  columns  carrying  moderate  loads, 
or  to  a  pitch  not  exceeding  3"  if  the  full  value  above  recommended 
is  to  be  used  in  considering  the  hooping.  A  reduction  of  the  allow- 
ance on  the  vertical  steel  should  be  made  where  the  spacing  of  the 
vertical  bars  exceeds  9"  center  to  center  in  proportion  to  fifty  percent 
of  the  increase  in  spacing  of  the  bars. 


SAFE    WORKING    STRESSES    IN    COLUMNS 


319 


6.  Safe  Ultimate  Limit  of  Compression.  A  wonderful  degree 
of  strength  which  may  be  developed  by  properly  hooped  and  longi- 
tudinally reinforced  concrete  columns  is  established  by  the  Considere 
experiments  and  it  becomes  a  question  as  to  how  great  values  it  is 
permissible  to  use.  Turner  has  found  it  permissible,  where  small 
diameters  are  desired,  to  use  as  the  approximate  limit  of  safe  design 
a  working  stress  as  high  as  4000  pounds  per  square  inch  of  cross 
section  of  the  core  including  the  vertical  steel.  Under  these  ulti- 
mate conditions  it  is  recommended  that  the  column  should  have 


Fig.  78.     Cut  showing  reinforcement  of  column  carrying  working  pressure  of  4000  pounds  per 

inch  on  the  core. 

enough  vertical  steel  to  carry  the  entire  load  at  little  more  than  the 
yield  point  value,  say  at  50,000  pounds  per  square  inch;  that  there 
be  sufficient  hooping  to  develop  the  value  of  the  load  figured  at 
40,000  pounds  per  square  inch  on  imaginary  verticals  corresponding 
to  2.4  times  the  volume  of  the  hooping;  and  that  the  gross  area  of 


H20  COACTION    OF    STEEL    AND    CONCRETE    IN    COLUMNS 

the  column  inside  the  fireproofing  should  be  sufficient  to  carry  the 
load  at  4000  pounds  per  square  inch. 

7.  The  Mode  of  Operation  of  the  Reinforcement  in  Concrete 
Columns.     In  a  practical  concrete  structure,  as  a  rule,  the  load 
is  brought  on  the  column  thru  the  beams  or  the  concrete  of  the  slab 
and  the  load  is  transferred  to  the  vertical  steel  thru  the  medium  of 
the  concrete  by  the  bond  between  the  concrete  and  the  steel. 

In  the  tests  by  Withey  and  those  of  some  other  American  inves- 
tigators, the  load  on  the  test  specimens  was  transferred  directly 
to  the  steel  by  direct  bearing  between  the  longitudinal  reinforce- 
ment and  the  face  plate  of  the  testing  machine.  This  makes  an 
important  difference  in  the  mode  of  operation  of  the  column.  The 
vertical  steel  is  a  more  rigid  element  under  compression  than  the  con- 
crete and  if  the  load  be  not  applied  directly  to  the  steel  but  be  brought 
on  the  steel  thru  the  surrounding  matrix,  the  coaction  of  the  two 
materials  is  brought  about  by  bond  shear  and  indirect  stress  gen- 
erated by  the  bond  shear  between  the  steel  and  the  concrete.  Now, 
since  there  is  no  slipping  of  the  steel  in  the  concrete  within  the  yield 
point  of  the  column,  the  same  amount  of  potential  energy  or  work 
of  deformation  is  stored  up  by  the  indirect  stress  of  the  bond  shear 
between  the  concrete  and  the  steel  as  is  stored  directly  in  the  steel. 
Hence  it  appears  that  in  this  casejthe  steel  by  help  of  bond  shear 
operates  to  store  energy  more  efficiently  up  to  the  point  where  cracks 
and  checks  occur  in  the  concrete  than  a  mere  comparison  of  the  rela- 
tive moduli  of  steel  and  concrete  would  indicate. 

This  phase  of  the  co-operation  of  the  steel  and  the  concrete  in 
the  column  requires,  for  its  greatest  efficiency,  a  strong,  rich,  con- 
crete, in  order  that  the  most  dependable  bond  may  be  secured. 

If,  however,  the  load  be  brought  upon  the  steel  by  a  rigid  bear- 
ing between  the  face  plate  of  the  testing  machine  and  the  steel  of 
the  longitudinal  reinforcement  and  in  like  manner  on  the  concrete 
core,  such  a  bearing  makes  the  elastic  deformation  of  the  steel  the 
same  as  that  of  the  concrete  thruout,  and  no  elastic  efficiency  by 
coaction  of  the  concrete  and  metal  exists,  such  as  occurs  in  the 
practical  structure. 

8.  Effect  of   Hooping.     The   crushing  of  solid  bodies   cannot 
take   place   without   lateral   swelling.     Therefore,    by  resisting  any 
such  swelling,  the  compressive  resistance  of  the  column  is  increased. 
In  the  practical  column  this  resistance  to  lateral  bulging  or  swelling 


STRESSES    IN    COLUMNS  321 

is  furnished  by  the  hooping  in  which  the  spacing  should  be  limited 
generally  to  four  inches  for  light  pressures  and  closer  spacing  for 
higher  pressures.  As  the  hooping  is  brought  into  play  by  lateral 
swelling  or  bulging,  the  degree  of  restraint  furnished  by  the  hooping 
alone  is  not  uniform  and  this  lack  of  uniformity  causes  the  concrete 
to  check  and  crack  outside  of  the  hooping  under  pressures  as  large 
approximately  as  the  ultimate  strength  of  plain  concrete. 

The  addition  of  vertical  steel  distributes  the  bulging  pressure 
from  band  to  band  or  hoop  to  hoop  to  such  an  extent  that  the  ver- 
tical steel  which  has  been  added  forms  beams  spanning  the  spaces 
between  coils  or  bands  and  does  so  to  an  extent  measured  by  the 
more  or  less  close  spacing  of  the  vertical  bars.  In  other  words, 
the  vertical  reinforcement  receives  lateral  pressures  between  the 
hoops  and  transfers  it  to  the  hoops  as  supports.  This  action  des- 
troys the  equilibrium  of  uniform  pressure  outward  upon  the  hoops 
and  tends  to  deform  the  configuration  of  the  circular  hoop  from 
circles  to  polygons  having  apices  at  the  point  of  bearing  of  the  ver- 
tical steel  against  the  hoops.  This  action  between  the  vertical  steel 
and  the  hooping  induces  indirect  stresses  between  the  hoops  and 
the  vertical  bars  similar  to  those  in  the  slab  with  two  way  reinforce- 
ment. 

The  outer  shell  of  the  concrete  is  subjected  to  direct  compression 
vertically  and  circular  tensions  horizontally,  brought  about  by  the 
bulging  tendency.  These  circular  tensions  are  reduced  between 
verticals  by  the  compression  brought  about  by  the  tendency  to 
change  configuration  in  the  hooping  just  mentioned,  while  the  ring- 
tension  opposite  verticals  is  increased  by  this  action.  The  lateral 
reinforcement  by  its  compressive  action  exerts  a  powerful  effect 
to  resist  bulging  and  to  prevent  circumferential  elongations  so  that 
the  reinforcement  (both  lateral  and  longitudinal),  enables  it  to 
withstand  much  greater  deformations  without  cracking  or  checking 
than  would  be  otherwise  possible. 

Let  us  review  the  action  of  the  column  as  the  load  is  applied  with 
reference  to  the  manner  in  which  the  potential  energy  of  internal 
work  is  stored  within  the  structure.  As  the  load  is  gradually  applied 
we  have  a  certain  elastic  deformation  and  the  internal  work  is  the 
mean  weight  times  the  deformation.  If  yielding  of  the  material 
occurs  or  scaling  of  the  shell,  a  certain  amount  of  energy  is  dissipated, 
equilibrium  is  destroyed,  and  new  energy  is  developed  by  downward 
motion  of  the  load  thru  increased  deformation  until  a  new  condition 
of  stability  of  equilibrium  in  established. 


322  TWO    KINDS    OF    COLUMNS 

Scaling  and  cracking  of  the  shell  means  a  loss  of  potential  energy 
stored  and  correspondingly  larger  deformations  which  are  inadmis- 
sible in  the  practical  structure.  The  addition  of  vertical  steel 
prevents  this  checking  and  scaling  and  dissipation  of  energy  because 
it  provides  a  storage  system  of  energy  which  is  stable  and  in  which 
the  storage  of  energy  by  indirect  stress  can  be  depended  upon. 

Coaction  between  the  hoops  and  the  vertical  steel  reduces  the 
deformation  and  hence  the  quantity  of  potential  energy  stored  for 
a  given  load  and  correspondingly  increases  the  efficiency  of  the 
structure  as  a  load  carrying  mechanism. 

9.  Comparison  of  Test  Data.  Having  pointed  out  in  detail, 
the  difference  in  action  between  these  two  kinds  of  columns,  it  is  in 
order  to  compare  test  data.  Column  Table  V,  gives  the  reinforce- 
ment, percentage  of  steel,  and  test  results  by  Withey*  of  a  series 
of  vertically  reinforced  and  hooped  columns  with  vertical  steel 
flush  with  the  ends  and  resting  against  bearing  plates.  These  may 
be  compared  with  the  tests  by  Bach,  and  the  tests  at  Phoenixville. 

Taking  up  the  comparison,  first,  with  the  series  by  Bach,  the  fol- 
lowing point  of  difference  is  noticeable.  The  test  results  obtained 
by  Withey  can  be  substantially  accounted  for  by  considering  the 
influence  of  the  concrete  and  vertical  steel  alone.  The  test  results 
by  Bach,  cannot  be  accounted  for  in  this  manner.  Both  series  show 
that  hooping  adds  toughness.  Both  indicate  that  larger  vertical  rods 
have  more  effect  in  raising  the  yield  point  than  rods  of  too  small 
diameter,  which  offer  little  resistance  to  lateral  bending.  The 
Bach  tests,  where  the  spiral  pitch  of  is  not  too  great,  and  where 
larger  vertical  rods  are  used  with  close  spacing  of  spirals,  are  in 
good  accordance  with  the  Considere  formula;  while  the  tests  by 
Withey  are  not  in  agreement  but  are  in  more  close  agreement  with 
a  different  formula,  which  considers  the  manner  in  which  the  load 
is  brought  upon  the  steel,  namely,  by  direct  pressure  instead  of 
by  bond  shear  as  in  usual  operation  of  the  column  in  practical 
building. 

Comparing  the  tests  by  Withey  with  those  at  Phoenixville,  the 
longitudinal  reinforcement  of  the  columns,  in  specimens,  J,  N,  P, 
0,  R,  and  Q,  is  substantially  the  same;  while  the  hooping  in  the 
Phoenixville  tests  was  much  greater.  The  increase  in  ultimate 
strength  in  the  Phoenixville  test,  for  one  percent  of  total  steel,  ranges 
from  two  to  three  times  as  much  as  in  the  tests  by  Withey.  This 

*Bulletin  of  the  University  of  Wisconsin,  No.  466. 


WITHEY'S  TESTS  OF  HOOPED  COLUMNS 


323 


COL.  TABLE  V 
Withey's  Tests  of  Hooped  Columns  Longitudinal  Steel  Full  Length  of  Shaft 


Reinforcement                 Percent 

Ulti- 

Stress 

Com- 

Reinforce- 

mate 

at  yield 

pressive 

Col. 

Round 
Vertical 

.                      ment           Core    Age 
Spiral                                      ar(>;i      in 

Mix 

strength 
lbs.-sq.in 

pt.  in 
core 

Pl/P 

strength 
of 

No. 

Rods 

Longi-               sq.  in.  days 

P/A 

lbs.-sq.in 

cylinders 

tudi-       La- 

PI  /A 

Ibs/sq.in 

No. 

Size 

Size 

Pitch 

nal        teral 

~T~ 

2           3 

4 

5           6            7:8 

9 

10 

11 

12 

13 

14 

C  1 

0          0 

i"  rd. 

1"            0        2.00 

78.5 

62 

-2-4 

4,660 

2,720 

0.58 

2,280 

C2 

0     |     0 

f" 

1"            0        2.00 

78.5 

62 

-2-4 

4,390 

2,330 

0.53 

2,190 

C3 

0     i     0 

1'      1       0     1  2.00 

78.5 

64 

-2-4 

3,660 

1,950 

0.53 

2,180 

C4 

0     i     0 

i' 

1' 

0        2.00 

78.5 

65 

-2-4 

3,410 

2,330 

0  .  68 

2,150 

Dl 

9     |     f" 

i  / 

1' 

3.50      2.00 

78.5 

58 

-2-4 

4,470 

3,290 

0.74 

2,150 

D2 

9          |" 

i' 

1' 

3.50      2.00 

78.5 

60 

-2-4 

4,200 

3,480 

0  .  83 

2,130 

D3 

9     !      1" 

i  ' 

1' 

3.50  !  2.00 

78.5 

61 

-2-4 

4,970 

3,860 

0.77 

2,380 

D4        9          1" 

i  ' 

1' 

3.50      2.00      78.5 

62 

-2-4 

5,360 

3,670 

0.68 

2,350 

HI        0 

0 

No.  7 

2' 

0        0.50      78.5 

57 

-2-3.5 

2,330 

1,950 

0.84 

2,040 

H2        00 

No.  7 

2' 

0        0.50      78.5!  57 

-2-3.5 

2,140 

1,760 

0.82 

*1,460 

G  1 

8 

\" 

No.  7 

2' 

2  .  00      0  .  50 

78.5 

49 

-2-3.5 

3,320 

2,710 

0  82 

2,100 

G2 

8 

I" 

No.  7 

2'         2.00      0.50      78.5 

58 

-2-3.5      3,280 

2,710 

0  .  83 

*  1,420 

I    1 

8 

W 

No.  7 

2' 

3.78  :  0.50      78.5 

57 

-2-3.5      4,240 

3,660 

0  87 

2,240 

I   2 

8 

W 

No.  7 

2' 

3.78      0.50      78.5 

57 

-2-3.5      4,080 

3,280 

0.80,     2,120 

J   1 

8 

I" 

No.  7 

2' 

6.11      0.50 

78.5 

58 

-2-3.5      5,190 

4,240 

0.82 

2,110 

J  2 

8 

I" 

No.  7 

O/ 

6.11      0.50      78.5 

58 

-2-3.5      5,050 

4,240 

0.84 

2,000 

L  1 

0 

0 

No.  7 

2' 

0        0.50      78.5 

57 

-2-3.5      2,680 

1,370 

0.51 

1,780 

L2 

0 

0 

No.  7 

2' 

0        0  .  50      78  .  5 

58 

-2-3.5      2,600 

1,370 

0.53      1,760 

Kl 

8 

k" 

No    7 

1' 

2.00  !   1.00 

78.5 

57 

-2-3.5 

4,050 

2,710 

0  .  67 

2,100 

K2 

8         :          -J-" 

No.  7 

1'         2.00  :   1.00      78.5 

57 

-2-3.5      3,760 

2,520 

0.67      1,890 

Nl 

8     i    W 

No.  7 

1' 

3.78  !   1.00 

78.5 

57 

-2-3.5      4,050 

3,470 

0.86 

1,880 

N2 

8         W 

No.  7 

1' 

3.78      1.00      78.5 

58 

-2-3.5      4,340 

3,280 

0.76 

1,720 

Ml 

8     ;     I" 

No.  7 

1' 

6.11      1.00 

78.5 

57 

-2-3.5 

4,790 

3,860 

0.811     1,660 

M2 

8     i     1" 

No.  7 

1' 

6.11      1.00 

78.5 

58 

-2-3.5 

4,580 

3,660 

0.80 

1,710 

P  1 

8          1" 

No.  7 

1' 

8.00      1.00 

78.5 

57 

-2-4 

6,760 

5,560 

0.821     2,380 

P2 

8          1" 

No.  7 

1' 

8  .  00      1  .  00 

78.5 

60 

-2-4 

7,090 

5,760 

0.81 

2,350 

01 

8 

i" 

i"  rd. 

1' 

6.11         .96 

78.5 

57 

-2-4 

6,510 

4,240 

0.65      2,270 

02 

8 

I" 

i//  « 

1' 

6.11         .96 

78.5 

57 

-2-4 

6,650 

4,620 

0.70 

2,690 

Rl 

8 

i" 

!"  " 

1' 

8.00        .96 

78.5 

53 

-2-4 

7,250 

5,000 

0.69)     2,310 

R2 

8 

i" 

i  //  « 

1'         8.00        .96 

78.5 

53 

-2-4 

6,680 

5,380 

0.80 

2,470 

Q  l 

8 

H" 

1"  " 

1'      110.12  i      .96 

78.5 

53 

-2-4 

6,190 

5,190 

0.84 

2,280 

Q  2 

8 

H" 

i  //  « 

1' 

10.12         .96      78.5 

53 

-2-4 

7,990 

6,340 

0.79 

2,330 

W  1 

0 

0 

6 

0 

0     i       0        86.6 

52 

-2-4 

2,660 

2,600 

W2 

(J 

0 

0 

0 

0     |       0        86.6 

52 

-2-4 

2,660 

2,400 

W3 

0 

0 

0 

0 

0     1       0 

86.6 

51 

1-2-4 

2,480 

2,250 

Note:     All  spirals  10"  in  diameter.     Cols.  C1-D4  are  120"-lg.     All  other  cols.  U)2"-lg. 


wide  discrepancy,  apparently  can  be  accounted  for  only  as  above 
outlined,  since  the  discrepancy  is  far  too  great  for  it  to  be  possible 
to  account  for  it  on  the  ground  of  difference  in  the  strength  of  the 
concrete.  The  concrete  in  the  test  of  the  Phoenixville  specimens 
is  undoubtedly  closely  comparable  to  the  concrete  of  the  specimens 
tested  by  Bach,  and  it  is  undoubtedly  a  somewhat  better  grade  than 
the  concrete  in  the  specimens  tested  by  Withey. 

The  conclusions  that  the  authors  draw  from  these  tests  are: 

That  it  is  bad  practice  to  splice  longitudinal  reinforcing  bars 
by  bearing  of  one  bar  upon  the  other. 

That  it  is  better  to  lap  bars  of  consecutive  columns  at  the  floor 
line  in  order  that  a  natural   adjustment    may    take    place    between 


324  CONCLUSIONS    AS    TO    REINFORCEMENT 

the  materials    as   the   load  is  brought  upon  the  concrete   during 
erection. 

That  a  column  constructed  by  lapping  the  bars,  so  that  the  natural 
relations  between  the  reinforcement  and  the  concrete  are  con- 
served, that  is,  that  indirect  stress  of  the  bond  shear  may  be  effective, 
substantially  doubles  the  efficiency  of  the  vertical  steel;  and  hence 
this  detail  of  reinforcing  should  be  used.  If  this  efficiency  is  to  be 
counted  upon,  the  diameter  of  the  vertical  bars  should,  for  like 
reason,  be  limited  preferably  to  1J  inch  diameter,  or  in  special  cases 
of  heavy  work  to  1J  inches  to  If  inch  round  bars. 

That  the  hooping  should  not  be  spaced  further  apart  than  four 
inches  for  light  pressures,  and  closer  spacing  should  be  used  for  heavy 
pressures. 

That  the  spacing  of  the  vertical  bars  should  not  exceed  nine 
inches  between  centers,  but  if  spaced  further  apart  their  efficiency 
should  be  considered  to  be  reduced  by  fifty  percent  of  the  relative 
increase  in  spacing. 

10.  Formula  for  Columns,  where  the  Load  is  Brought  upon 
the  Steel  by  Direct  Bearing  on  Metal.  Tests  of  this  type  of  column 
show,  up  to  the  yield  point,  so-called,  that  the  effect  of  the  hooping 
is  relatively  small;  that  beyond  the  yield  point  the  hooping  is  brought 
into  play,  giving  the  column  a  degree  of  toughness  and  ability  to 
stand  increased  load  without  sudden  failure,  altho  accompanied 
by  considerable  deformation.  Neglecting  the  circumferential  rein- 
forcement and  the  concrete  outside  of  it, 

Let  A    =  area  of  core  occupied  by  concrete  and  longitudinal  rein- 
forcement having  a  steel  ratio  of  p,  so  that 

As  =  pA  =  cross  section  of  longitudinal  steel. 

Ac  =  (1  —  p)  A  =  cross  section  of  concrete  core. 

W  =  total  load  to  yield  point  that  column  would  carry  without 

circumferential  steel. 
ei   =  .00125"  =    average    deformation    per    unit    of    length    at 

yield  point  of  column. 

W  =  (As  Es  +  Ac  #CH.     Let  ES/EC  =  n,  then 
w  =  W/A  =  Ese,[  (1  —  p)/n  +  p] 
is  the  load  at  yield  point  per  unit  of  cross  section  of  concrete. 

Let  Es  =  30,000,000,  and  n  =  15;  then 

w  =  37,500  [  (1  —  p)/15  +  p'  ]  =  2500  +  35,000  p 


FORMULA    FOR   WITHEY's    COLUMNS  325 

The  point  here  designated  on  the  diagram  of  unit  deformations 
and  load  is  at  intersection  P  of  the  tangents  to  the  two  slopes  of 
the  curve,  Fig.  78. 

Equation  (1)  is  in  good  agreement  with 
the  series  of  tests  made   by   Professor   M. 
O.  Withey,  C.   E.,    in    1910,    at   the    Uni- 
versity   of   Wisconsin,*    with    longitudinal 
reinforcement  from  1   percent    to    10    per- 
cent    and     circumferential      reinforcement 
from  0.5  percent  to  2  percent,    given    in 
Col.    Table    V.        Professor    Withey     has 
written    empirical     formulas     for     several 
different  percentages  of  circumferential  re- 
Fi<r  78     Diagram  showing        inforcement  which  differ  slightly  from  this 
and  from  each  other;  but  they  do  not   re- 
fer precisely  to  the  point  P    as  above   defined,    but   to   a   slightly 
smaller  value  of  W.     The  point  P  does  not  actually  fall  on  the  test 
curve  of  unit  deformations  and  loads,  but  is  so  related  to  it  geometri- 
cally as  to  make  its  use  advisable.     The  actual  deformation  at  this 
load  will  be  somewhat  in  excess  of  the  assumed  value  of  el. 

On  page  49,  in  Bulletin  466,  University  of  Wisconsin,  the  follow- 
ing conclusions  are  drawn  from  the  results  of  these  tests  regarding 
the  behavior  of  columns  with  rigid  metallic  bearing  for  the  longitud- 
inal reinforcement  built  of  concrete  similar  to  those  tested  by  Withey. 

1.  "  Closely  spaced  spiral  reinforcement  will  greatly  increase 
the  toughness  and  will  considerably  increase  the  ultimate  strength 
of  a  concrete  column,  but  will  not  materially  affect  the  yield  point. 
The  ultimate  strength  under  dead  load  will  doubtless  be  some- 
what less  than  the  value  obtained  from  a  testing  machine.     Since 
spiral  reinforcement  should  be  employed  principally  as  a  factor 
of  safety  against  sudden  collapse,  more  than  one  percent  does  not 
appear  to  be  necessary.     On  account  of  lack  of  stiffness  in  columns 
made  from  this  grade  of  concrete  and  reinforced  with  spirals  only, 
it  seems  necessary  to  use  some  longitudinal  steel. 

2.  "Longitudinal  steel  in  combination  with  such  spiral  rein- 
forcement raises  the  yield  point  and  ultimate  strength  of  a  column 
and  increases  its  stiffness.     As  was  shown  in   Bulletin  No.   300 
columns  reinforced  with  longitudinal  steel  only  are  brittle  and 
fail  suddenly  when  the  yield  point  of  the  steel  is  reached;  but 
they  are  considerably  stronger  than  plain  concrete  columns  made 
from  the  same  grade  of  concrete. 

3.  "From  behavior  under  test  of  the  columns  reinforced  with 
spirals  and  vertical  steel  and  the  results  computed,  it  would  seem 
that  a  static  load  equal  to  30  to  40  percent  of  the  yield  point  would 
be  a  safe  working  load." 


*Bulletin  No.  466. 


326  WORKING    STRESSES    FOR    COLUMNS 

These  tests  are  conclusive  as  regards  the  deportment  of  the  column 
without  vertical  steel.  Spiral  reinforcement  gives  a  large  increase 
in  the  ultimate  strength  and  while  providing  safety  against  sudden 
collapse,  does  not  increase  the  yield  point  at  which  the  fireproofing 
commences  to  scale,  and  hence  does  not  permit  material  increase 
in  the  working  stress  over  that  of  the  vertically  reinforced  column 
types  with  ordinary  ties. 

1 1 .  Working  Stresses,  (a)  The  case  of  no  direct  metallic  bear- 
ings. In  the  vertically  reinforced  and  hooped  column,  in  which  the 
hooping  follows  closely  the  percentage  of  the  vertical  steel  recom- 
mended previously,  namely,  .35  to  .45  of  the  volume  of  the  verticals, 
and  the  verticals  are  not  less  than  three  quarters  of  an  inch  in 
diameter,  and  spaced  approximately  as  recommended,  the  yield 
point  of  the  column  may  be  taken  as  eighty  percent  of  the  ultimate 
strength.  This  would  give  a  yield  point  value,  for  a  column  to 
which  the  Considere  formula  is  applicable  of  .80  x  1.5  =  1.2  times 
the  crushing  strength  of  the  concrete  times  the  core  area,  plus 
eighty  percent  of  the  yield  point  value  of  the  steel  in  verticals  and 
hoops. 

Since  a  column  of  this  character  may  be  depended  upon  with 
certainty,  if  ordinary  care  is  used  in  erecting  the  work,  the  factor 
of  safety  of  2.5  may  be  employed.  This  would  give,  in  round 
numbers,  800  pounds  per  square  inch  for  the  concrete  of  the  core, 
12,000  pounds  per  inch  on  the  verticals,  and  16,000  pounds  per  square 
inch  on  the  hooping,  treated  as  imaginary  verticals,  for  the 
working  stress,  the  hooping  being  assumed  to  be  a  drawn  wire,  and 
possessing  a  higher  yield  point  than  the  material  of  the  verticals. 

With  a  1  :  If  :  3  mix,  twenty  percent  higher  values  may  be 
assigned  to  the  core. 

(6)  Metallic  bearings.  Where  vertical  reinforcement  is  used, 
and  the  load  is  brought  on  the  vertical  steel  by  bearing  on  metal, 
there  is  much  less  certainty  of  uniformity  of  joint  action  between 
the  matrix  and  the  reinforcement  than  where  the  longitudinal  bars 
are  lapped;  and  hence  a  somewhat  higher  margin  of  safety  should 
be  allowed.  Taking  this  margin  of  safety  as  the  reciprocal  of  3f, 
the  writers  would  recommend  a  working  stress  not  exceeding  700 
pounds  per  square  inch  on  the  concrete  plus  10,000  pounds  per  square 
inch  on  the  vertical  steel  for  this  type  of  column  with  rigid  end 
bearing,  limiting  the  percentage  of  hooping  to  not  less  than  one 
half  of  one  percent  nor  less  than  twenty  percent  of  the  vertical  steel, 
and  the  vertical  steel  limited  to  not  more  than  eight  percent  of  the 
column  area.  This  factor  of  safety  is  based  on  the  yield  point  value 
of  the  column. 


STRUCTURAL    STEEL    AND    CONCRETE    COLUMNS  327 

12.  Structural  Columns  Filled  with  Concrete.     As  steel  is  more 
rigid  than  concrete,  the  higher  the  percentage  of  the  steel  the  less 
the  assistance  we  may  expect  the  concrete  to  be  to  the  steel,  and 
while  the  formulas  given  heretofore  for  the  column  in  which  the 
vertical  steel  is  loaded  in  large  part  by  direct  bearing  of  metal  on 
metal  agrees  with  tests  closely,  it  should  be  noted  that  the  per- 
centage  of  steel   in  these   specimens  was  not   high.     Accordingly, 
it  would  seem  conservative  to  reduce  the  allowance  on  concrete  in 
cases  where  the  percentage  of  steel  exceeds  eight  percent  of  the 
combined  area  of  the  concrete  and  steel  in  proportion  to  the  in- 
crease in  steel  above  this  percentage. 

13.  Concrete  Columns  Compared  to  Structural  Steel.     From 

the  Phoenixville  tests,  it  would  seem  that  there  is  a  higher  degree  of 
uniformity  in  the  tested  strength  of  reinforced  concrete  columns 
made  with  ordinary  care  and  with  well  designed  reinforcement 
than  with  average  structural  steel  columns.  The  reason  is 
that  in  the  concrete  column,  we  have  a  solid  core.  The  larger  the 
column  the  greater  the  strength,  in  strong  contrast  with  some  struc- 
tural steel  columns.  In  the  structural  steel  columns,  we  have  the 
uncertainty  due  to  the  form  and  make-up  of  the  section.  In  the 
make-up  of  most  small  struts  and  columns  constructed  of  steel  and 
iron,  we  have  a  specific  ratio  of  the  area  of  the  web  and  flanges  con- 
sisting of  channels,  or  in  other  forms,  a  complete  circular  or  box 
section,  such  as  the  Phoenix  column  or  the  box  made  up  of  two  chan- 
nels and  two  plates.  The  semi-empirical  formula  for  struts  has 
been  worked  out  for  the  radius  of  gyration  of  those  forms  of  sections 
which  are  comparable  in  the  case  of  laced  channel  struts  to  a  certain 
distribution  of  metal  between  the  web  and  the  flange  and  a  certain 
ratio  of  flange  width  to  depth,  which  ratios,  under  standard  sections, 
are  comparable  to  the  box  and  Phoenix  sections.  When  there  is  a 
wide  variation  from  these  proportions,  the  formula  based  on  the 
radius  of  gyration  is  inapplicable  as  is  proved  in  the  original  design 
of  the  columns  of  the  Quebec  Bridge.  In  this  case  the  area  of  the 
flange  as  compared  to  the  web  was  about  one  tenth  of  this  standard 
proportion  between  the  area  of  the  web  and  the  channel,  and  there 
is  no  experimental  data  in  existence  covering  the  action  of  the  lat- 
ticed bars  and  secondary  stress  for  such  a  combination. 

In  concrete  work  such  uncertainties  are  eliminated  by  the 
uniform  solid  core.  With  ordinary  care  there  can  be  no  doubt 
about  securing  a  solid  casting,  if  the  type  of  reinforcement  selected 
is  that  recommended  in  this  chapter,  namely,  one  in  which  there  is 


328  WALL    COLUMNS 

no  obstruction  whatever  in  the  core  to  prevent  the  concrete  from 
flowing  freely  and  filling  the  same  completely  and  so  securing  a 
solid  casting.  Improper  handling  of  the  concrete  might  lead  to 
considerable  reduction  of  the  strength  of  the  section,  but  the  damage 
which  might  occur  thru  bad  workmanship  is  if  anything  much  less 
in  concrete  work  than  in  steel  construction  and  the  confidence  in  the 
builder  in  the  integrity  of  his  concrete  work  should  be  correspond- 
ingly greater. 

14.  Wall  Columns  and  Interior  Columns  in  Skeleton  Con= 
struction.  Wall  columns  differ  from  the  interior  column  in  that 
the  load  comes  to  them  from  one  side  instead  of  equally  or  approx- 
imately equally  from  all  directions  as  it  does  under  a  uniform  load 
on  the  floors. 

Whether  the  floor  is  beam  and  girder  construction  or  flat  slab 
construction  supported  directly  upon  columns  in  view  of  the  mono- 
lithic connection  between  the  floor  and  the  columns  certain  eccen- 
tric loads  or  bending  stresses  are  brought  upon  the  wall  columns  to 
a  greater  extent  than  is  the  case  with  interior  columns. 

The  amount  of  bending  induced  in  the  column  will  depend,  first 
on  the  rigidity  of  the  floor,  and  second,  upon  whether  the  resistance 
of  the  floor  to  deformation  is  furnished  by  beam  action  or  circum- 
ferential slab  action.  If  the  resistance  is  by  circumferential  slab 
action,  the  effect  upon  the  column  is  far  less  for  the  same  deflection 
than  it  would  be  in  case  of  beam  action,  because  the  slab  tends  to 
twist  from  all  directions  and  so  in  a  large  measure  the  effect  of  the 
load  in  producing  bending  and  deformation  of  the  column  is  counter 
balanced  by  the  loads  coming  to  the  column  from  the  sides  instead 
of  being  augmented  by  these  loads  so  much  as  is  the  case  in  the  beam 
action  of  one-way  reinforcement. 

Favoring  Conditions.  The  walls  of  a  building  are  made  vertical 
and  as  the  number  of  stories  increases,  the  columns  are  reduced  in 
diameter,  but  kept  flush  or  vertical  on  the  outer  face.  This  produces 
an  eccentric  application  of  the  column  load  from  the  upper  stories 
upon  the  columns  of  the  lower  stories  which  in  some  large  measure 
off-sets  and  holds  in  equilibrium  the  bending  moment  of  the  floor 
loads. 

Little  attention  is  usually  paid  to  this  difference  in  conditions. 
The  wall  column  is  commonly  made  approximately  the  same  as  the 
interior  column  except  in  the  upper  stories  where  it  is  possible  to 


TEMPERATURE    EFFECTS  329 

carry  the  wall  columns  up  for  the  last  three  stories  without  reduction 
in  size,  making  thereby  a  provision  for  the  columns  of  these  upper 
stories  which  would  be  most  affected  by  the  eccentric  bending 
moment  referred  to. 

Good  practice  would  limit  the  dimensions  of  columns  to  sixteen 
inches  as  a  minimum  size  in  office  buildings  and  in  warehouse  con- 
struction, eighteen  inches  as  a  minimum. 

Provision  of  material  in  the  beam  to  take  the  entire  bending 
resistance  figured  as  simply  supported  at  the  wall  column  and  con- 
tinuous over  the  interior  columns  will  in  no  wise  eliminate  this  bend- 
ing stress  in  the  column  itself,  and  does  not  excuse  failure  to  make 
such  provision. 

In  fact  if  the  beams  are  treated  as  beams  fixed  at  the  inner  ends 
and  free  at  the  outer  end,  and  the  five  rod  type  of  reinforcement 
used  in  the  Turner  beam  is  adopted,  the  positive  moment  provided 
for  near  mid  span  is  W  L/14.4  nearly,  or  .Q7W  L,  while  the  max- 
imum positive  moment  in  the  beam  freely  supported  and  continuous 
over  a  number  of  spans  in  the  end  span  requires  a  coefficient  of  from 
0.07  for  two  spans  to  .08  for  three  spans,  and  .077  or  .78  for  a  greater 
number,  so  that  in  this  type  of  design  there  is  nearly  sufficient  pro- 
vision without  considering  the  stiffness  of  the  wall  column.  Never- 
theless, in  the  view  of  the  authors  the  provision  recommended  should 
be  followed  in  practice. 

Bending  from  Wind  Pressure.  In  view  of  the  monolithic  charac- 
ter of  reinforced  concrete  construction,  wind  has  much  less  effect  on 
a  structure  of  this  kind  than  is  the  case  with  the  steel  frame,  since 
each  floor  forms  a  monolith  of  enormous  lateral  rigidity  causing 
columns  all  to  act  in  perfect  unison.  Where  the  building  is  narrow 
the  columns  should  be  fully  spliced  at  the  floor  level  which  doubles 
the  resistance  of  the  column  to  flexure.  Where  the  buildings  are 
broad  and  only  six  to  eight  stories  in  height,  provision  of  this  charac- 
ter is  unnecessary  except  in  wall  columns  which  should  be  well 
spliced. 

15.  Temperature  Effect  on  Columns.  Changes  in  temperature 
cause  expansion  and  contraction  of  the  concrete  floor.  This  is 
taken  up  by  bending  in  the  column  and  the  in  and  out  motion  of 
the  walls  where  they  are  above  the  ground  line.  There  is,  however, 
a  tendency  for  the  basement  walls  to  crack  every  thirty  or  forty 
feet  because  of  the  restraint  of  the  surrounding  earth.  There  is  a 
further  tendency  for  brick  walls  to  crack  where  there  is  expansion 


330  ECONOMIC    DESIGN    OF    COLUMNS 

if  brick  piers  are  used  in  place  of  well  reinforced  concrete  piers  in  a 
long  building.  In  a  building  over  300  feet  in  length,  no  dependence 
should  be  placed  upon  either  a  plain  brick  or  a  plain  concrete  pier 
without  reinforcement  to  withstand  temperature  stresses  of  expan- 
sion and  contraction  of  the  floors  since  unsightly  cracks  and  checks 
will  very  likely  occur  from  this  cause.  In  a  building  over  300  feet 
in  length  either  a  full,  well  reinforced  concrete  skeleton  should  be 
provided  or  if  bearing  walls  are  used  the  building  should  be  cut  at 
250  foot  intervals  so  that  it  will  allow  expansion  and  contraction 
without  damage  to  the  brick  work  or  undue  strain  on  the  concrete  of 
the  floors  and  columns. 

16.  Economic  Column  Design.  With  reinforcing  steel  in  the 
column  figured  at  2J  cents  a  pound  and  concrete  at  $6.00  per  cubic 
yard,  we  may  estimate  the  proportion  of  steel  and  concrete  needed 
to  obtain  the  most  economic  support  from  the  standpoint  of  cost. 
Since  a  cubic  foot  of  concrete  weighs  150  pouds  and  a  cubic  foot 
of  steel  485  pounds,  a  cube  of  concrete  weighing  one  pound  would 
have  a  volume  3.2  times  as  great  as  a  cube  of  steel  of  the  same 
weight,  and  have  a  face  nearly  one  and  one-half  times  as  great.  On 
this  basis,  for  equal  cost  in  carrying  load,  fs/fc  =  37,  but  in  a  fairly 
rich  concrete  n  =  10  to  15,  so  that  a  load  can  be  carried  with  less 
cost  on  a  concrete  pier  than  upon  one  of  reinforced  concrete,  but  in 
carrying  the  load  in  this  manner,  the  diameter  of  the  columns  are 
greater,  which  is  objectionable  in  that  they  occupy  valuable  room. 
Unbalanced  bids  are  sometimes  received  on  this  basis.  One  puts  in  a 
bid  using  a  column  of  a  small  diameter  as  specified,  thus  using  more 
steel,  but  saving  space;  another  makes  a  column  larger  by  two  or 
three  inches  and  uses  less  steel  and  puts  in  a  lower  price.  The 
owner  is  frequently  persuaded  to  accept  the  large  columns  without 
giving  the  regular  bidder  an  opportunity  to  revise  his  bid  on  the  same 
basis. 

It  should  be  noted  that  the  efficiency  of  the  hooping  varies  as  the 
diameter  of  the  core,  while  the  core  area  varies  as  the  square  of  the 
diameter;  hence  a  small  increase  in  the  diameter  of  the  core  permits 
a  large  decrease  in  the  amount  of  hooping  and  vertical  steel.  Hoop- 
ing costing  1.6  times  as  much  as  the  vertical  bars  in  a  column  in  place 
has  an  efficiency  2.4  times  that  of  the  same  cost  of  metal  in  vertical 
steel.  On  the  other  hand,  certain  relative  proportions  between  the 
amount  of  the  vertical  steel  and  hooping  are  necessary  to  secure 
the  best  results,  and  these  proportions  should  be  adhered  to  if  the 
formulas  recommended  are  to  be  applied. 


331 


CHAPTER  IX 
FOUNDATIONS 

1.  Bearing  Value  on  Soil.     In  a  building  the  floor  loads  are 
carried  to  the  columns, or  to  the  walls  in  case  bearing  walls  are  used, 
and  the  weight  is  concentrated  on  small  areas  of  ground  at  the  foot- 
ings of  the  columns  and  walls.     Evidently  if  we  are  to  avoid  settle- 
ment the  weight  must  be  distributed  over  a  sufficient  area.     The 
following   are   suggestions  for   safe   loading  for   foundations  where 
the  material  can  be  clearly  defined. 

Ordinary  ledge  rock,  such  as  good  shale,  limestone  and  the  like, 
twenty  to  thirty  tons  per  square  foot.  Granite,  trap  where  the  ledge 
is  not  shattered,  fifty  to  seventy  tons  per  square  foot.  Hard  pan, 
seven  tons  per  square  foot.  Gravel,  five  tons.  Clean,  coarse  sand, 
four  tons.  Fine  sand,  with  a  little  clay,  three  to  three  and  one-half 
tons.  Hard  clay,  three  tons.  Clay,  such  as  is  to  be  found  in  Regina, 
which  is  rather  soft,  not  over  one  and  one-half  tons.  Blue  clay 
of  Winnipeg,  two  to  two  and  one-half  tons. 

In  each  case,  however,  it  is  well  for  the  engineer  to  look  into  the 
conditions  carefully  unless  thoroly  conversant  with  the  locality. 
His  judgment  as  to  the  bearing  power  of  the  soil  should  be  checked, 
if  doing  business  in  a  strange  city,  by  a  careful  examination  of  the 
buildings  resting  on  similar  foundation  and  general  inquiry  among 
experienced  members  of  his  profession.  This  caution  may  prove 
of  value  to  the  engineer  doing  business  over  an  extended  area, 
particularly  if  he  is  acting  in  a  consulting  capacity  for  a  contracting 
firm  assuming  responsibility  for  the  design. 

In  cases  where  there  is  filled  ground,  marsh,  quick-sand  and 
the  like  it  is  frequently  necessary  to  use  a  pile  foundation  or  distribute 
the  weight  over  the  entire  area.  In  general  in  reinforced  concrete 
construction,  it  is  economical  to  use  a  comparatively  thin  footing 
and  thoroly  reinforce  it  and  make  the  concrete  a  rich  mix. 

2.  Column  Footings  and  Method  of  Figuring.     Fig.  80  shows 
a  form  of  footing  which  seems  somewhat  objectionable  for  the  reason 
that  as  usually  placed  the  concrete  is  worked  rather  dry  in  order  to 


332 


COLUMN    FOOTINGS 


make  the  slope  without  top  forms,  and  the  contractor  does  not 
ordinarily  get  it  in  place  so  that  it  can  be  depended  upon  with 
certainty. 

A  preferable  form  is  shown  in  Fig.  81,  in  which  the  footing  is 


Fig.  80. 


made  in  two  layers.  The  bottom  layer  is  cast  with  the  rods  at  the 
bottom,  then  the  column  assembled  and  the  top  layer  cast  with  the 
column.  Its  advantage  from  a  practical  standpoint  is  first,  that 


1= 


Fig.  81. 


the  upper  layer  of  rods  assist  the  footing  in  resisting  the  shearing 
strain  of  the  column  which  tends  to  punch  thru  the  footing  and  secondly 
it  also  assists  in  distributing  the  load  out  over  the  lower  plate.  The 


PILE    FOUNDATIONS  333 

bottom  plate  being  cast  with  a  wet  mix  we  can  depend  with  certainty 
on  the  bond  between  the  steel  and  the  concrete  by  the  shrinkage 
of  the  wet  mixture  as  we  could  not  in  the  footing  first  mentioned. 

In  computing  the  lower  plate,  the  cross  sectional  area  of  the  steel 
should  be  such  as  to  provide  for  a  bending  moment  of  the  upward 
pressure  on  the  under  side  of  the  plate  distributed  uniformly  over 
it  with  the  column  as  the  point  of  support. 

The  arm  of  the  total  pressure  either  side  of  the  center  may  be 
taken  roughly  as  five-eighths  of  the  half  diameter  of  the  footing,  hence 
the  actual  bending  moment  is  5/16  Wb.*  The  metal  acting  as  a 
flat  plate  would  be  as  we  have  shown  for  the  square  panel  twice  as 
efficient  as  single  way  reinforcement  and  as  the  sectional  area  of 
each  bar  comes  into  play  on  each  side  of  the  center  of  the  footing 
we  can  use  four  times  the  area  of  all  rods  crossing  the  footing  times 
.85  of  the  total  thickness  of  the  two  layers  times  the  working  stress 
on  the  steel  as  the  resisting  moment. 

The  amount  of  steel  placed  in  the  top  layer  is  more  a  matter  of 
practice  than  that  of  exact  computation.  The  rods  are  generally 
made  the  same  size  as  those  used  in  the  lower  layer  and  no  time 
is  wasted  in  computing  the  stresses  thereon.  It  is  preferable  to 
take  long  rods  and  bend  them  hair-pin  style  for  the  footings  rather 
than  to  use  short  rods. 

There  is  a  further  advantage  in  ordering  the  rods  this  way,  that 
in  case  the  steel  for  the  footings  is  delayed  stock  steel  can  be  used 
while  the  steel  ordered  for  footings  may  be  used  elsewhere  in  the 
building  later. 

3.  Pile  Foundations.      Where  piles  if  used  will  be  continually 
wet  and  there  is  no  possibility  of   changing  conditions   from   that 
to  alternate  drying    out  and  wetting,  there  is  no  type  of  reinforced 
concrete  pile  that  can  compete  with  timber  piles.     However,  where 
the  piles  are  liable  to  be  above  the  permanent  water  line  or  it  becomes 
necessary  to  excavate  thereto,  then  and  there  concrete  piles  become 
an  economic  method  of  building  up  the  foundation. 

Practical  concrete  piles  may  be  divided  into  two  classes,  one, 
those  which  are  made  up  first  and  driven  afterwards,  and  second, 
those  in  which  a  core  or  form  is  used  and  the  hole  filled  in  with 
concrete  reinforced  or  otherwise. 

4.  Driven  Piles.     Hennebique  was  one  of  the  first  to  use  rein- 
forced concrete  piles  that  were  made  up  separately  and  then  driven. 


In  this  notation  b  is  the  breadth  or  diameter  of  footing. 


334 


PILES    MADE    UP    AND    DRIVEN 


In  design  they  were  similar  to  his  columns  which  have  been  pre- 
viously illustrated,  with  longitudinal  reinforcement  coupled  with 
either  of  several  arrangements  of  lateral  ties. 

Considere  in  his  pile  design,  shown  in  the  accompanying  Fig.  82, 
commonly  made  use  of  spirals  for  lateral  reinforcement  in  place  of 
Hennebique's  ties.  His  form  of  piles  is  used  to  a  large  extent  on 
the  continent  and  is  similar  as  regards  the  point  at  its  lower  extremity 
to  Hennebique's. 

In  driving,  it  is  customary  to  use  a  water  jet  for  loosening  up  the 
sand  or  earth  at  the  bottom  and  rap  or  jar  the  pile  into  place  with  a 


Fig.  82  Showing  Reinforcement  of  the  Considere  pile. 

hammer.  A  steam  hammer  should  be  used  in  driving  concrete  piles 
rather  than  a  drop  hammer,  since  with  the  heavy  ram  and  short 
stroke  of  the  steam  hammer  there  is  less  shock. 

5.  Piles  Cast  in  Place.  Coming  under  the  second  class  are 
the  Raymond  and  Simplex  piles. 

The  Raymond  piles  are  made  tapering  to  secure  greater  resis- 
tance against  settling.  The  manner  of  constructing  Raymond 
concrete  piles  is  as  follows:  A  steel  pile-core  the  size  and  shape  of 
pile  desired  is  encased  in  a  thin,  closely  fitting  shell.  The  core  and 


PILES    CAST    IN    PLACE 


335 


shell  are  then  driven  into  the  ground  by  means  of  a  pile-driver  in 
the  usual  manner.  By  a  simple  and  ingenious  device  the  core  is 
collapsed  or  shrunken  slightly,  so  that  it  loses  contact  with  the  shell, 
and  is  easily  withdrawn,  leaving  in  the  ground  a  clean,  perfectly 
formed  hollow  tube  of  the  size  and  depth  required,  which  has  only 
to  be  filled  in  with  the  best  Portland  cement  concrete  to  complete 
the  pile. 

Fig.  83,  shows  the  Raymond  pile  expanded  ready  to  be  driven. 


Fig.  83.     Raymond  Pile   Core.      This  core  is  used  to  drive  to  rock  and  tapers  from 
20  to  13  inches.      The  cut  shows  core  full  size  as  it  is  driven. 


336 


THE    RAYMOND    PILE 


Fig.  84,  shows  the  core  in  the  leaders  with  the  shell  on  the  right. 


Fig.  84.     Pile  Core  Collapsed  or  Shrunken,  and  partly  withdrawn  from  the  shell. 

The  core  is  expanded  when  driven  and  collapsed  to  be  withdrawn 
from  the  shell. 

The  relative  merits  of  the  various  kinds  of  piles  would  apparently 
depend  on  outside  conditions.  A  pile  like  the  Raymond  pile  should 
be  most  suitable  for  a  clay  soil  where  the  consistency  or  cohesiveness 
of  the  clay  is  such  that  the  pile  core  can  be  driven  and  the  shell 
omitted. 


SIMPLEX    PILES 


337 


Raymond  piles  are  made  of  various  lengths,  tapering  generally 
from  20"  at  the  top  to  6"  at  the  botton,  making  a  symmetrical  cone 
affording  material  resistance  to  soil  penetration  by  friction. 

For  the  Simplex  pile  a  casing  is  first  driven,  and  then  as  the  casing 
is  pulled  up,  concrete  is  deposited  and  rammed  in  place,  forcing  it 
out  to  a  somewhat  greater  diameter  than  the  shell  that  has  been 
driven. 

Figs.  85  and  86,  show  the  make-up  of  the  Simplex  pile. 
With  the  steel  casing  is  driven  a  point,  either  of  steel  or  concrete, 
and  afterwards  the  shell  is  gradually  withdrawn  and  the  hole  filled 
with  concrete  as  the  shell  is  filled  up. 


Fig.  86  Fig.  85 

Simplex  Pile. 

For  use  in  earth  that  is  reasonably  firm  in  its  texture  and  free 
from  water,  the  preparatory  removable  pile,  (see  Fig.  85)  is  used. 
This  pile  form  consists  of  a  length  of  extra  heavy  wrought  iron  pipe, 
fitted  with  a  suitable  driving  head  of  oak,  and  a  conical  steel  point 
of  a  somewhat  larger  diameter  than  the  pipe,  and  fitted  with  an 
automatic  air  valve.  This  preparatory  tube  is  driven  into  the  ground 
to  the  required  depth,  and  then  withdrawn  without  difficulty,  and 
the  hole  so  produced  is  filled  with  well  rammed  concrete.  This 
form  of  pile  can  be  constructed  of  any  desired  length,  as  the  pre- 
paratory tube  can  be  driven  and  removed  with  but  a  fraction  of 
the  force  required  in  the  planting  or  removal  of  the  ordinary  pile. 
It  can  also  be  driven  thru  ground  of  a  density  quite  impenetrable 
by  any  wooden  pile  and  to  almost  any  desired  depth,  as  there  is  no 
appreciable  frictional  resistance,  as  the  depth  increases,  either  in 
driving  or  withdrawing  the  tube. 


338 


SIMPLEX    PILES 


The  ramming  process  forces  the  larger  pieces  of  the  aggregate 
into  the  sides  of  the  hole,  materially  adding  to  the  frictional  hold 
of  the  pile  on  all  parts  of  its  surface. 

Where  the  earth  is  soft,  marshy,  or  where  quicksand  or  water 
is  encountered,  a  detachable  " point"  of  concrete,  (see  Fig.  87)  is 
substituted  for  the  fixed  one  of  steel.  This  concrete  point  is  driven 
to  the  required  depth,  and  as  the  pipe  is  being  lifted  off,  concrete 
is  gradually  filled  in  and  rammed  home  thru  the  pipe,  care  being 
taken  that  a  head  of  the  concrete  be  maintained  inside  of  the  pipe 
while  it  is  being  thus  gradually  withdrawn.  By  this  system  all 
water  is  displaced  and  the  possibility  of  the  sides  of  the  aperture 
closing  in  is  entirely  removed. 


Fig.  87.     Concrete  Points  for  Simplex  Piles. 


Corrugated  piles  have  been  patented  by  Frank  B.  Gilbreth. 
One  of  the  claims  advanced  in  their  favor  is  that  the  corrugations 
assist  in  jetting  the  piles  into  place. 


CORRUGATED    PILES  339 

Fig.   89  shows  a  driver  handling  one  of  the  corrugated  piles. 


Fig.  89.      Driving  Corrugated  Concrete  Piles. 

Fig  90  shows  the  cushion  cap  used  in  driving  the  pile. 


Cl  Cap 


rior.  Section  A-B.  Wooden  Buffer   X 

90.     Cushion  Cap  used  in  driving  the  corrugated  concrete  pile. 


340 


PILE    FOUNDATIONS 


Concrete  piles  have  this  advantage  over  wood  piles;  they  do 
not  decay,  are  not  subject  to  destruction  by  insects  and  furnish 
a  durable  foundation  regardless  of  soil  conditions;  they  can  be  used 
in  dry  filled  ground  as  well  as  in  wet  soil,  which  may  dry  out  and 
cause  wood  piles  to  decay. 

A  pile  foundation  generally  tends  to  prevent  settlement  by  the 
packing  and  settling  of  the  soil  in  the  vicinity  of  the  pile,  frequently 
rendering  the  bearing  value  of  the  soil  around  the  piles  materially 
greater.  See  Fig.  91. 


91.     A  Trench  Filled  with  Concrete  Piles. 


ADVANTAGE  OF  CONCRETE  OVER  WOOD  PILES 


341 


Fig.  92,  gives  a  fair  idea  as  to  the  advantage  of  a  reinforced  con- 
crete pile  foundation  over  the  wood  pile  which  must  be  cut  off  in 
the  vicinity  of  the  low  water. 


•3S  7on$  per 


Reinforced  Concrete  Piles.  Wood  Piles. 

Fig.  92.     Showing  the  advantage  of  concrete  over 
wood  pile  foundations. 

6.  Safe  Bearing  Loads  for  Piles.  Two  cases  are  to  be  dis- 
tinguished; that  of  piles,  the  lower  end  of  which  rests  upon  a  hard 
strata  and  that  of  the  ordinary  pile  which  is  supported  largely  by 
skin  friction  of  the  material  into  which  it  is  driven.  The  capacity 
of  the  former  is  determined  by  the  strength  of  the  pile  as  a  column 
thru  the  upper  or  soft  strata  while  the  bearing  power  of  the  latter 
is  some  function  of  the  penetration  under  a  given  drop  of  a  ram  of 
a  given  weight. 


342  FORMULAS    FOR    SAFE    BEARING    OF    PILES 

Many  formulas  have  been  proposed,  but  the  only  formulas  in 
anything  like  general  use  are  known  as  the  Engineering  News 
Formulas.  They  are: 

For  a  pile  driven  with  a  drop  hammer   P  =  — 

8+1 

For  a  pile  driven  with  a  steam  hammer  P  =  ~ 

S+0.1 

to  which  P  is  the  safe  load  in  pounds,  W  the  weight  of  the  hammer 
in  pounds,  h  the  fall  of  the  hammer  in  feet,  and  S  the  penetration 
or  sinking  in  inches  under  the  last  blow,  assumed  to  be  at  an  approx- 
imately uniform  rate.  They  are  deduced  for  wood  piles;  but  are 
the  best  there  are  for  concrete  piles. 

Morsch  gives  the  Brix  formula  as  usually  employed  on  the 
Continent  : 

hQ2g 
2e  (Q+gf 

wherein  h  is  the  fall  of  the  hammer; 
Q  the  weight  of  the  hammer; 
g  the  weight  of  the  pile; 

e  the  penetration  of  the  pile  under  the  last  blow; 
p  is  double  the  safe  allowable  load  for  the  pile. 

The  quantity  e  will  naturally  be  the  average  of  the  last  few  blows. 


343 


CHAPTER  X 

ELEMENTS  OF  ECONOMIC  CONSTRUCTION  AND  COST  OF 
REINFORCED  CONCRETE  WORK 

1.  Introductory.     Reinforced  concrete  construction  of  buildings 
presents  problems  which  from  an  economical  standpoint  are  so  com- 
plicated that  they  cannot  well  be  investigated  as  questions  of  max- 
ima and  minimia  by  means  of  equations  which  show  the  cost  of  the 
several  variable  items  and  determine  their  proper  relation  by  mathe- 
matical treatment.     For  this  reason  it  is  necessary  to  pursue  the 
investigation  along  simpler  lines,  by  taking  up  questions  of  general 
arrangement,  column  spacing,  spacing  of  beams,  choice  of  type  of 
reinforcement,  cost  of  centering  for  each  kind,  cost  of  aggregate,  etc. 

2.  Column  Spacing.     Eighteen  feet  center  to  center  of  columns 
in  each  direction  usually  costs  less  than  for  shorter  spans  while  the 
increase  in  cost  with  increase  of  column  spacing  up  to  twenty  feet 
is  very  little,  if  the  loads  are  heavy,  and  the  building  high.     In  whole- 
sale hardware  buildings,  the  customary  requirement  is  that  of  lines 
of  shelving  and  boxing  about  twelve  feet  centers.     For  such  a  build- 
ing, diagonal  spacing  of  columns  with  flat  slab  construction,  making 
the   columns   about   seventeen   feet   between   centers,   will   provide 
the  desired  spacing  in  a  multiple  of  twelve  feet  for  the  boxing  and 
shelving,  i.  e.  twenty-four  feet  in  directions  parallel  to  the  sides  of 
the  building. 

3.  Floors.     In  treating  the  strength  of  floors  by  comparison 
on  the  principle  of  proportion  it  has  been  noted  that  the  coefficient 
of  bending  with  continuous  flat  slab  type  is  smaller  than  the  corres- 
ponding coefficient  for  beam  construction.     This  advantage  is  off- 
set for  light  loads  by  the  disadvantage  of  the  smallness  of  depth 
which  is  a  factor  in  the  moment  of  resistance.     Further,  stiffness 
varies  as  the  square  of  the  depth,  but  the  coefficient  for  deflection 
is  smaller  than  for  simple  beams.     Hence  it  is  evident  that  for  stiff- 
ness under  light  loads,  lower  percentages  of  steel  are  requisite  than 
in  the  beam  type  of  construction.     Where  the  loads  are  heavy  the 
continuous  flat  slab  type  becomes  more  and  more  economical  com- 
pared with  other  types  of  floor  as  the  depth  of  slab  increases  by  reason 


344  SLAB    FLOORS    COMPARED 

of  the  relatively  small  coefficient  of  bending.  A  little  computation 
as  to  the  cost  of  concrete  and  steel  shows  that  considerable  variation 
from  the  exact  economical  proportions  of  the  two  materials  for  a 
given  strength  will  make  comparatively  little  difference  in  the  ulti- 
mate cost,  whereas  in  the  flat  slab  types  using  a  smaller  amount  of 
steel  and  more  concrete  does  make  a  large  increase  in  the  stiffness. 

Flat  slab  types  have  the  advantage  of  simpler  centering  and  lower 
cost  of  placing  the  steel,  but  aside  from  that  they  possess  little 
advantage  over  the  two-way  beam  system  for  light  loads,  tho  requir- 
ing less  material  both  of  concrete  and  steel  for  heavy  loads  and  mod- 
erate spans  up  to  twenty-five  or  thirty  feet,  and  they  compare 
favorably  for  greater  spans  where  the  loads  exceed  five  hundred 
pounds  per  square  foot. 

Slabs  reinforced  in  one  direction,  being  supported  on  two  sides 
only,  are  at  a  disadvantage,  since  the  coefficient  of  bending  is  three 
times  as  high  with  one-way  reinforcement,  as  it  is  for  the  slab  rein- 
forced in  two  ways  and  supported  on  four  sides. 

In  comparing  Types  III  and  IV,  it  should  be  observed  that 
if  the  beams  of  Type  III  are  of  twice  the  thickness  of  the  slab  of 
Type  IV,  the  same  weight  of  reinforcement  is  required  in  the  beams 
of  Type  III  that  is  required  in  the  whole  floor  of  Type  IV. 

Usual  proportions  would  be  a  depth  of  beam  equal  to  three  times 
the  thickness  of  the  slab  of  Type  IV  requiring  two  thirds  the  steel 
for  the  beams. 

The  apparent  moment  to  be  resisted  in  the  slab  supported  on 
four  sides  Type  III  is  but  two  thirds  that  to  be  borne  by  slab  of  Type 
IV.  Hence  a  slab  may  be  used  of  less  thickness  than  with  Type 
IV  tending  to  equalize  the  concrete  quantities  and  leaving  the  dif- 
ference in  cost  of  forms  and  steel  to  be  considered. 

Type  II  is  frequently  made  of  a  combination  of  hollow  tiles  and 
thin  reinforced  concrete  beams  between  the  hollow  tiles,  in  order  to 
secure  economic  depth  and  reduced  weight  for  long  spans,  with 
decreased  deflection  under  load.  Where  this  type  is  used,  the 
cost  of  centering  is  kept  down  to  a  figure  approaching  that  of  the 
flat  slab  types  but  extra  care  is  required  in  placing  the  steel  and  the 
labor  of  putting  tile  in  position  and  the  cost  of  the  tile  is  added. 
Where  tile  is  low  in  cost  and  concrete  material  is  high  in  cost,  tile 
and  concrete  for  a  light  building  may  be  more  economical.  On 
the  other  hand,  the  risk  of  erection  is  greater  with  this  type  on  ac- 
count of  the  brittleness  of  tile  and  the  character  of  failure  of  one 


CENTERING  345 

way  reinforced  slabs  as  pointed  out  in  the  discussion  of  beams. 
For  heavy  loads,  however,  this  type  cannot  compete  with  the  natural 
concrete  types,  III  and  IV. 

4.  Centering.  As  we  have  noted,  centering  is  one  of  the 
important  considerations  since  the  cost  of  centering  runs  to  twenty- 
five  to  thirty-five  percent  of  the  total  cost  of  the  ordinary  floor. 

In  the  selection  of  the  type  of  floor  to  be  used,  the  cost  of  center- 
ing is  so  large  an  item  that  it  should  be  given  careful  consideration 
in  any  approximate  solution  of  an  economic  design. 

Where  the  spans  are  long  and  the  loads  are  light  and  the  cost  of 
concrete  materials,  stone,  gravel  and  cement  are  very  high,  flat 
slab  types  cannot  compete  with  the  beam  type.  Type  III  as  a 
beam  and  slab  type  is  more  economical  than  other  beam  types  for 
heavy  loads  and  panels  of  sixteen  to  eighteen  feet.  Where  the 
columns  are  equally  spaced,  the  economy  is  greatest.  Where  the 
spacing  is  such  that  the  panels  are  rectangular  and  one  side  is  less 
than  six-tenths  of  the  longer  side,  its  economy  disappears  and  it 
is  preferable  to  use  an  intermediate  beam,  thus  dividing  the  slab 
into  panels  more  nearly  square. 

For  joist  and  girder  construction  surmounted  by  light  slabs,  the 
spacing  of  the  ribs  is  governed  by  the  character  of  the  centering 
used.  If  the  centering  is  arranged  in  panels  so  that  it  is  easily 
handled  as  such,  six  to  ten  foot  spacing  of  beam  joists  works  out 
very  economically.  Considerable  increase,  however  in  the  cost  of 
centering  is  brought  about  by  the  additional  framing  of  the  joists 
to  the  main  beams  or  girders. 

No  exact  rule  can  be  given  covering  all  cases,  as  the  conditions 
of  the  problem,  such  as  size  of  the  building,  manner  in  which  it 
divides  up  for  the  purposes  intended,  etc.,  fix  so  many  of  the  condi- 
tions that  any  rule  disregarding  these  conditions  might  readily  be 
misleading.  The  number  of  stories  figures  largely  in  the  cost  of 
centering.  The  cost  of  framing  beam  boxes  as  a  rule  must  be 
figured  for  three  stories.  Additional  stories,  if  the  framing  is  worked 
out  so  that  the  same  boxes  can  be  used  over  and  over  again  reduces 
the  cost  per  foot  greatly  and  these  are  conditions  which  must  be 
taken  into  consideration  in  any  practical  comparison  of  different 
types. 

5.  Columns.  The  hooped  column  is  the  safest  type  to  erect 
and  where  the  loads  are  at  all  heavy  the  most  economical  type  to 
adopt.  The  economy  of  the  reinforcement  depends  largely  upon  the 


346  BEARING    WALL    OR    SKELETON 

adoption  of  the  proper  proportion  of  hooping  and  vertical  steel. 
The  hooping  adds  to  the  toughness  of  the  column  and  to  its  ultimate 
strength  but  does  not  raise  the  point  where  the  column  shell  com- 
mences to  scale  and  chip  unless  the  hooping  be  combined  with  the  pro- 
per proportion  of  vertical  steel.  When  this  has  been  done,  very  high 
values  indeed  can  be  safely  developed.  The  carrying  capacity, 
however,  of  the  column  is  secured  at  a  minimum  cost  by  concrete 
rather  than  by  reinforcement.  Still  a  certain  amount  of  steel  is 
necessary  to  secure  toughness  and  resist  flexure.  Again,  considera- 
tion of  the  value  of  floor  space  frequently  limits  the  size  of  the  column 
which  the  architect  or  owner  is  willing  the  designer  should  employ. 

As  to  what  may  be  done  with  the  reinforced  concrete  column, 
Turner  has  used  a  twenty-seven  inch  core,  heavily  banded  and  re- 
inforced vertically,  for  working  loads  from  eleven  to  twelve  hundred 
tons.  This  pressure  being  a  little  over  4000  pounds  per  square  inch 
of  core  area.  The  development  of  such  high  working  values  by  heavy 
reinforcement,  tho  unobjectionable  from  the  standpoint  of  safety 
and  desirable  from  the  standpoint  of  occupying  small  floor  space 
is  not  economical  from  the  standpoint  of  first  cost  of  providing  a 
post  of  proper  capacity  for  a  given  load.  A  rich  concrete  is  more 
economical  than  a  lean  mixture,  and  where  the  loads  are  heavy  a 
1  :  1^  :3  mixture  is  recommended  in  place  of  the  usual  1:2:4 
employed  under  ordinary  conditions. 

6.  Bearing    Walls   or    Full    Concrete    Skeleton.     A    very    im- 
portant question  in  economic  design  is  the  question  as  to  whether 
bearing  walls  are  to  be  preferred  to  a  full  concrete  skeleton. 

For  such  a  low  building  as  four  to  five  stories,  bearing  walls  are 
generally  cheaper  than  a  full  concrete  skeleton.  For  buildings 
higher  than  five  stories  a  full  concrete  skeleton  with  curtain  walls 
costs  less  than  heavy  bearing  walls. 

In  putting  up  a  concrete  skeleton,  the  additional  cost  involved 
in  making  provision  for  two  or  three  additional  stories  is  generally 
so  small  that  it  is  advisable  for  the  owner  to  make  this  provision 
if  there  is  a  reasonable  probability  that  he  may  use  the  additional 
floor  space  in  the  future  and  the  value  of  the  real  estate  warrants 
such  investment. 

7.  Concrete,  or  Brick  Exterior  Walls.     In  some  cases  where 
the  concrete  aggregate  is  cheap  and  union  brick  layers'  wages  are 
high  it  is  better  to  use  a  concrete  exterior  wall.     Generally,  however, 
exterior  walls  may  be  constructed  much  more  cheaply  of  brick,  or 


SELECTION    OF    AGGREGATE  347 

of  some  material  that  can  be  laid  up  without  the  necessity  of  using 
forms,  since  the  forms  for  exterior  wall  construction  run  into  money 
quite  rapidly. 

8.  Rich  Mixture.     Economic  construction  in  reinforced  concrete 
requires  a  rich  mixture.     This  is  a  necessity,  first,  from  the  stand- 
point of  certainty  of  computation,  second  from  the  standpoint  of 
quick  hardening  which  enables  the  early  removal  of  the  forms  with 
economy  incident  to  repeated  use  of  the  same  form  lumber,  and 
from  the  standpoint  of  economy  due  to  the  fact  that  we  can  use  less 
material  of  good  quality  which  we  can  absolutely  depend  upon  than 
we  can  of  material  of  an  inferior  quality  and  uncertain  character 
which  is  liable  to  be  discredited  by  reason  of  its  slow  hardening 
through  the  lack  of  necessary  amount  of  cement.     Further,  where 
the  material  used  has  been  of  good  quality  the  construction  can  be 
increased  in  strength  to  any  desired  degree  by  the  addition  of  more 
good  concrete  tho  the  strength  so  secured  will  not  be  at  so  low  a 
cost  as  if  the  original  design  was  for  heavier  construction. 

9.  Economy  in  Selecting  Aggregate.     Good  bank  gravel  when 
obtainable  makes  an  excellent  aggregate.     Its  adaptability  for  the 
purpose  should  be  determined  by  screening  out  the  sand  and  pebbles 
which  are  under  J  inch  diameter  and  comparing  the  volume  of  sand 
with  the  volume  of  coarse  aggregate.     If  these  proportions  depart 
from  those  desired,  that  is,  one  cement,  two  sand,  and  four  of  the 
coarse  aggregate,  then  the  cement  content  must  be  increased  to  make 
the  mortar  of  the  proper  proportion,  or  else  crushed  stone  must  be 
added  to  the  mixture. 

Where  the  expense  of  securing  crushed  stone  is  high  and  the  cost 
of  cement  is  low,  the  addition  of  more  cement  to  keep  the  mortar 
a  true  1  :  2  mix  is  cheaper  than  the  addition  of  stone,  and  in  a  way 
preferable,  because  with  the  excess  of  mortar  there  is  less  liability 
of  voids  and  poor  work.  Sometimes  crushed  stone  and  gravel  is 
not  available  and  a  good  hard  furnace  or  smelter  slag  may  be  secured. 
Slag  should  be  examined  for  chemical  impurities  which  might  injure 
the  cement  and  for  hardness  which  determines  its  fitness  as  a  good 
aggregate. 

10.  Cinders.     Cinders   are    sometimes    used    as    an   aggregate 
for   concrete.     Cinder  from  the   soft   Iowa   coal  is  generally  very 
injurious  to  the  cement.     In  fact  it  may  be  stated  as  a  general  rule 
that  the   only   cinder   fit   to   make   a   permanent  concrete    is    that 
which  is  a  hard  or  more  or  less  vitrified  clinker  such  as  generally 
results  from  burning  soft  coal  with  a  mechanical  stoker.     Too  great 


348  ADAPTABILITY    OF    CONCRETE    CONSTRUCTION 

care  cannot  be  exercised  in  this  respect  as  upon  the  character  of  the 
aggregate  and  its  freedom  from  sulphur  or  other  injurious  chemical 
elements  which  would  damage  the  cement,  depends  the  permanence 
and  integrity  of  the  work. 

In  general  a  clinker  concrete  should  not  be  used  where  a  high  de- 
gree of  strength  is  required.  It  is  desirable  to  use  it  for  such  work  as 
roof  work  where  the  spans  are  short  and  it  is  desired  to  nail  a  tile 
or  slate  cover  to  the  concrete  roof  slab.  For  such  purposes  the 
concrete  should  not  be  mixed  too  rich,  otherwise  it  will  be  difficult 
to  nail  into  it. 

11.  Adaptability.  Reinforced  concrete  is  not  adapted  for  long 
spans  and  light  loads.  For  instance,  in  a  span  of  fifty  or  sixty 
feet  in  a  shop  or  factory  building,  having  only  a  light  roof  load, 
reinforced  concrete  is  not  an  economical  material  to  use.  Struc- 
tural steel  costs  far  less  and  is  generally  employed.  For  bridges  of 
long  spans  and  light  loads  reinforced  concrete  is  not  economical  in 
first  cost.  Where  the  loads  are  heavy,  as  for  a  city  bridge,  and  the 
grade  such  that  there  is  opportunity  for  ample  rise,  a  reinforced 
concrete  arch  may  be  built  at  a  cost  not  greatly  exceeding  that  of  a 
good  structural  steel  bridge  and  when  the  maintenance  charges 
are  considered  the  concrete  will  be  the  least  expensive. 

For  office  buildings  and  ordinary  business  blocks  reinforced 
concrete  will  in  general  save  the  owner  from  one-half  to  three-quarters 
of  the  cost  of  a  structural  steel  skeleton. 

Reinforced  concrete  is  particularly  well  adapted  for  school 
buildings.  The  difference  in  cost  between  the  ordinary  timber 
floor  and  a  reinforced  concrete  floor  will  frequently  not  exceed  ten 
or  fifteen  cents  per  square  foot.  With  this  fact  in  mind  it  is  really 
astonishing  to  observe  how  frequently  dangerous  fire  traps  are 
erected  to  serve  as  school  buildings  on  which  expensive  and  orna- 
mental exteriors  have  been  used  when  plainer  buildings  with  fire- 
proof construction  could  be  erected  for  the  same  money. 

The  architect  for  a  school  building,  in  order  to  make  a  show, 
frequently  specifies  a  fancy  brick  exterior,  terra  cotta  or  stone 
trimmings  and  other  external  frills  and  then  economizes  in  the 
interior  construction  of  the  building  by  the  use  of  timber  joists 
l|  or  If  inches  thick  by  16"  covered  with  f  inch  rough  floor  and 
|  inch  hardwood  finished  floor,  with  wood  lath  and  plaster  on  the 
under  side,  electric  wires,  heating  flues,  etc.,  between  the  joists. 
This  is  a  construction  in  which  if  a  fire  once  started  there  would  be 


STIFFNESS    UNDER   IMPACT  349 

hardly  time  for  the   occupants  of  the   building  to  escape    before 
the  collapse  of  the  floor  with  the  probable  loss  of  life  incident  thereto. 

The  Collinwood  disaster  well  illustrates  this  fact.  The  state  of 
Wisconsin  has  passed  a  law  making  it  compulsory  to  build  school 
buildings  of  fireproof  materials  and  other  states  may  well  follow  her 
example. 

For  buildings  subject  to  the  vibration  of  heavy  machinery 
concrete  steel  construction  has  many  advantages.  Properly  designed, 
the  joints  (connections  of  floor  to  columns)  are  far  more  rigid  than 
in  any  of  the  old  types  of  construction,  hence  a  rigid  building  costs 
least  in  concrete  steel. 

The  Forman-Ford  Company,  plate  glass  dealers,  etc.,  make 
the  statement  that  twenty-five  percent  more  work  is  done  in  their 
cutting  and  polishing  department  in  their  new  concrete  building 
with  the  same  men  than  in  the  old  timber  framed  structure 
previously  occupied  owing  to  the  increase  in  rigidity  of  the  structure. 

Impact  or  shock  at  any  point  of  a  steel  structure  is  propagated 
longitudinally  along  elastic  members  extending  in  a  linear  direction 
from  the  point  and  it  goes  practically  undiminished  to  the  far  ends 
of  these  members  where  it  is  subdivided  among  other  members  and 
propagated  still  further.  An  allowance  up  to  eighty  or  ninety  per- 
cent is  usually  added  for  impact  to  the  static  effect  of  a  moving  load 
in  bridges. 

Impact,  or  the  dynamic  effect  upon  any  point  of  a  reinforced 
concrete  slab,  however,  is  entirely  different  from  this.  In  the  first 
place,  the  effect  of  the  blow  does  not  travel  in  one  direction  only 
but  in  all  directions  radially  from  its  point  of  application,  so  that 
in  a  very  thin  slab  its  effect  at  any  other  point  would  be  inversely 
as  the  distance  and  in  a  very  thick  slab  inversely  as  the  square  of 
the  distance.  This  would  make  the  allowance  for  impact  in  the 
thick  concrete  floor  of  a  bridge  or  building  very  small  in  comparison 
with  that  inevitable  in  steel  construction. 

Secondly,  the  effect  of  impact  must  be  inversely  proportioned 
to  the  weight  of  the  body  receiving  the  blow.  Now  in  monolithic 
concrete  construction  the  mass  affected  is  of  far  greater  weight  than 
would  be  the  case  in  a  steel  frame  which  is  made  up  of  independent 
steel  members  and  tile  as  in  the  old  style  buildings  and  for  that 
reason  the  effect  of  impact  on  a  monolithic  concrete  floor  would  be 
reduced  by  a  large  percentage. 

Third,  the  continuity  and  stiffness  of  the  floor  greatly  reduces 


350  RESISTANCE    OF    REINFORCED    CONCRETE    TO    SHOCK 

its  vertical,  lateral  and  torsional  deformations  below  those  of  the 
steel  structure.  The  work  done  during  an  impact  and  its  effect 
depend  on  the  amplitudes  of  the  deformations.  In  particular  the 
horizontal  resistance  of  the  slab  is  many  thousand  times  that  of  the 
steel  members  in  a  structure.  The  vibratory  energy  absorbed  by 
the  slab  during  impact  is  consequently  small. 

Fourth,  the  small  amount  of  energy  which  is  absorbed  is  not 
transmitted  (as  it  is  in  a  highly  elastic  and  resilient  structure)  to  a 
considerable  distance  in  the  slab,  but  owing  to  the  nature  of  the 
concrete,  is  dissipated  near  its  source,  transformed  into  heat,  and 
rapidly  absorbed. 

Fifth,  concrete  slabs  are  tough  and  not  brittle,  like  terra  cotta, 
for  example,  so  that,  in  cases  where  great  weights  have  fallen  on  them, 
little  effect  has  been  produced,  whereas  brittle  slabs  such  as  those 
of  concrete  and  tile  have  been  smashed  under  such  circumstances, 
and  have  failed. 

The  concrete  of  the  compression  zone  is  such  a  shock-absorber 
as  to  protect  the  tension  zone  from  jarring  and  vibration,  both  as 
regards  steel,  in  tension  and  the  bond  of  the  concrete  to  the  steel  as 
well. 

For  all  these  reasons  the  shock  which  a  rolling  load  imparts  to 
a  slab  is  inconsiderable,  and  is  absorbed  and  dissipated  so  readily 
that  it  is  a  negligible  factor,  rendering  reinforced  concrete  adaptable 
for  use  in  railroad  structures  and  in  buildings  where  the  service 
is  most  severe  from  shock  of  machinery  such  as  beater  floors  in  paper 
mills  and  hammers  in  stamping  and  working  metal. 

As  illustrating  these  statements,  an  interesting  accident  occurred 
in  Winnipeg  where  a  heavy  cornice  block  of  hard  Tyndale  lime- 
stone, weighing  between  a  quarter  and  a  half  ton,  was  dropped 
seventy  feet,  striking  a  5J  inch  concrete  slab  19  feet  square  in  the 
center.  The  effect  of  this  blow  was  a  small  dent  half  inch  deep  where 
the  corner  of  the  stone  struck  the  slab  and  the  stone  itself  was 
badly  broken  and  shattered.  The  slab,  however,  was  uninjured 
and  showed  no  cracks  or  evidence  of  over-strain. 

In  Minneapolis,  a  steel  water-tower  failed  under  wind  pressure 
and  a  fifty  ton  tank  dropped  ten  feet  on  a  light  roof  slab  of  concrete 
reinforced  after  the  manner  of  the  Mushroom  system  in  four  direc- 
tions. The  slab  was  approximately  22  feet  by  23  feet  clear  span, 
seven  inches  thick,  reinforced  with  7  /16  inch  rounds  averaging 
eight  inches  centers  in  two  directions  with  diagonal  belts  of  fourteen 


KAl'IIMTY    OF    ERECTION  3.51 

7/16  inch  rounds.  The  shock  of  the  fifty  tons  falling  ten  feet  dented 
the  slab  down  about  eight  inches  and  produced  some  shear  cracks 
but  it  carried  the  load  and  caused  the  contents  of  the  tank  to  fall 
outside  of  the  building,  demolishing  two  freight  cars  and  the  awn- 
ing over  the  loading  platform.  The  toughness  of  this  slab  saved 
a  wreck  of  the  side  of  the  building  and  the  elevator  machinery. 

Other  interesting  cases  might  be  cited,  but  it  is  thought  that  these 
unusual  examples  illustrate  the  remarkable  toughness  and  dependa- 
bility of  concrete  reinforced  in  multiple  directions  and  its  great 
adaptability  from  the  standpoint  of  capacity  to  withstand  severe 
usage  with  high  degree  of  safety  for  all  purposes  when  scientifically 
designed  and  properly  executed. 

12.  Rapidity  of  Erection  and  Ease  of  Securing  Material.  No 
type  of  building  can  be  so  rapidly  or  quickly  designed,  detailed  and 
erected,  as  the  natural  types  of  reinforced  concrete  construction. 
If  we  take  Type  IV,  for  instance,  a  single  computation  is  sufficient 
for  a  panel  and  where  panels  are  tabulated  for  various  loads  two 
hours'  work  is  sufficient  to  make  the  computation  for  a  given  size 
of  factory  building  or  manufacturing  plant,  including  an  estimate 
of  the  cost  of  reinforcement,  quantities  of  concrete  and  centering 
with  sufficient  precision  for  bidding  purposes. 

In  no  type  of  building  construction  can  the  materials  be  secured 
so  promptly  as  for  the  reinforced  concrete  structure.  An  ordinary 
four  or  five-story  building  can  sometimes  be  erected  complete  in 
the  time  ordinarily  required  to  get  out  the  shop  details  for  a  struc- 
tural steel  frame.  Especially  where  the  building  is  irregular  in  form 
there  is  this  advantage  with  reinforced  concrete,  that  the  joints  are 
made  with  the  cement  in  plastic  form,  that  the  rods  can  be  lapped 
more  or  less  over  the  supports,  avoiding  the  necessity  for  the  large 
amount  of  figuring  required  for  the  skew  connections  of  structural 
work,  hence  the  engineer's  end  in  this  line  of  building  construction 
is  greatly  simplified. 

The  accompanying  figure  shows  the  rapidity  of  construction  of 
the  Bostwick  Braun  Building;  its  condition  August  1st  showing  the 
sea  wall  and  adjacent  footings  incomplete;  November  llth,  showing 
the  centering  nearly  completed  for  1he  roof,  or  eight  stories  in  place. 

This  is  merely  a  usual  example  in  the  construction  of  a  large 
building.  A  floor  per  week  of  the  concrete  skeleton  and  rough 
slab  can  easily  be  erected  under  favorable  weather  conditions. 

The  is  little  difference  between  a  large  building  and  a  small  build- 
ing in  thin  respect,  since  with  a  larger  building  it  is  possible  to  rig  up  in 


BOBTWICK-BRAUN  BUILDING 


\u".  1st,,  Showing  S«-;iw:ill  and  Adjiuwnl.  Footings  Incomplete. 


Nov.  lllli,  ShowiiiK  (ViitnriiiK  Nearly  Complete  for  tho  Hoof,  10i«ht  StorioM  in  I'liioo. 


ANALYSIS    OK    COST 


a  manner  that-  will  facilitate  the  handling  the  work  more  rapidly. 
In  fact,  where  there  is  a  large  area  it  permits  the  employment  of  more 
men  and  makes  it  possible  to  keep  them  at  work  on  the  various 
features  such  as  centering,  pouring  concrete,  placing  steel,  in  one 
continuous  operation,  centering  going  ahead  for  one  floor  as  the 
steel  is  being  placed,  the  concrete  men  in  turn  following  up  those 
placing  the  steel.  When  the  carpenters  are  thru  with  this  floor  they 
immediately  proceed  to  erect  the  forms  for  the  next  on  thut  portion 
of  the  floor  where  the  concrete  has  been  cast. 

Where  the  amount  of  work  to  be  handled  runs  eight  to  ten  thou- 
sand yards,  it,  pays  well  to  rig  up  with  overhead  bins  and  mixing 
plant  and  to  arrange  for  the  use  of  half-yard  dump  cars  in  placing 
the  concrete.  Where  the  yardage  is  much  less  the  wheelbarrow 
or  two-wheeled  truck  and  scale  hoist  becomes  an  economic  method 
of  handling  material.  The  later  mixers  are  arranged  with  a  charging 
device  which  saves  wheeling  material  up  an  incline  as  was  customarily 
done  in  earlier  work. 

13.  Analysis  of  Items  of  Cost.  In  arriving  at  a  detail  estimate 
of  cost  we  have  the  following  items  to  consider: 

B 0,818  of  Labor 
Materials  and  cost  of  handling: 

( Vment 

Sand 

Stone 

Water 

( 'ommon  labor 

( !ost  of  plant 

( 'ost  of  metal 

( 'ost,  of  unloading 

Labor,  cost/  of  bending  (union  or  common; 

Labor,  cost  of  pl.-icing  (union  or  common) 

( <ost  of  lumber 

Cost  of  framing  beam  boxes,  columns,  etc. 

(/ost  of  erecting  and  rehandling 

Slab  forms,  beam  forms  and  column  forms. 


( 'oncrctc, 
Unit  price 


Stee 


Centering 


>r  gravel 


>Quantities  and  base  prices. 


Season  of  the  year. 

Floor  finish  or  strip  fill. 

Dead  expense. 

(leneral  data  on  costs  per  foot,  of  floor  and  items  entering  into  it 


354  QUANTITIES    AND    UNIT    PRICES 

14.  Labor,  Unit  Prices,  Quantities  of  Material.  Under  the 
general  heading  of  labor  the  contractor  must  consider,  first,  the 
wages  per  hour;  second,  the  character  and  efficiency  of  the  labor., 
whether  the  labor  is  union  or  non-union,  probability  of  strikes  and 
delay  of  work  into  the  unfavorable  season  when  artificial  heat  must 
be  used. 

Where  trade  unions  are  strong  the  specialist  in  reinforced  con- 
crete can  never  tell  when  his  work  will  be  tied  up  by  some  disagreement 
between  master  plumbers  and  walking  delegates  or  other  trades 
with  which  he  has  no  relation  whatever  further  than  that  a  sym- 
pathetic strike  may  be  called  without  notice  or  grievance  at  any 
time  and  his  operations  brought  to  a  standstill. 

This  condition  means  idle  equipment  and  sometimes  cost  of 
heating  materials  and  may  mean  readily  an  additional  cost  of  five 
to  ten  per  cent  in  the  work. 

In  Chicago  the  bricklayers'  union  demands  that  the  contractor 
shall  keep  employed  an  extra  brick  foreman  who  is  supposed  merely 
to  watch  the  placing  of  concrete  wherever  there  may  be  brick  work 
on  the  job. 

While  a  labor  union  should  prove  of  benefit  to  employer  and 
employee  alike  in  case  its  motto  is  efficiency  and  skilled  service, 
it  loses  both  public  sympathy  and  support  when  it  cultivates  ineffi- 
ciency and  loads  the  work  up  with  men  who  are  useless  as  is  the  case 
with  an  extra  foreman.  In  many  cases  concrete  is  used  for  exterior 
walls  in  place  of  brick  where,  were  it  not  for  this  short-sighted 
policy,  brick  would  be  used  from  an  economic  standpoint. 

For  purpose  of  discussion  and  comparison  we  will  take  the 
following  costs:  Common  labor,  $2.25  per  day  of  ten  hours;  car- 
penters, $3.00  per  day  of  eight  hours;  steel  to  be  placed  by  common 
labor  at  $2.25  to  $3.00  per  day. 

Unit  Price  of  Concrete 
1:2:4  Mix 

Material  for  one  cubic  yard  of  concrete,  wet  mix : 

Cement 1 J  bbls. 

Crushed  stone 9    cubic  yards. 

Sand 45  cubic  yards. 

WThere  a  crusher  run,  including  dust  of  good  hard  crystalline 
stone  is  used  the  sSnd  may  be  readily  reduced  to  one-third  yard. 


COST    OF    HANDLING  35f) 

The  above  amounts  required  to  make  a  cubic  yard  of  wet  mixed 
concrete  in  place  may  vary  somewhat  on  the  character  of  the  crushed 
stone  or  gravel,  but  for  estimating  purposes  they  are  conservative. 

1  :lj  :3Mix 
Material  for  one  cubic  yard  of  concrete,  wet  mix: 

Cement 2  bbls. 

Sand 43  cubic  yards. 

Stone 85  cubic  yards. 

1:3:5  Mix 
Material  for  one  cubic  yard  of  concrete,  wet  mix: 

Cement 4j  sacks  =  1.125  bbls. 

Sand 52  cubic  yards. 

Stone 85  cubic  yards. 

Labor  of  Handling 

Given  an  ordinary  equipment  such  as  a  half-yard  Smith,  Cube 
or  Ransome  machine  the  labor  cost  of  handling  concrete  may  be 
stated  as  follows: 

Wheelbarrow  gang,  from  $1.25  to  $1.50  per  cubic  yard,  including 
cost  of  coal  or  gasoline  for  the  engine.  In  walls,  footings  or  where 
there  is  quite  a  mass  this  may  be  reduced  to  $1.00  per  cubic  yard. 

On  a  large  job  where  one-half  yard  cars  are  used  and  overhead 
bins  for  handling  the  aggregate  by  gravity,  the  labor  cost  may  be 
reduced  to  35c  to  40c  per  cubic  yard.  To  this  must  be  added, 
however,  the  cost  of  fitting  up  the  plant  which  will  increase  this 
figure  to  sixty  or  even  seventy  cents  per  cubic  yard. 

The  labor  costs  will  increase  or  decrease  as  the  price  of  common 
labor  is  above  or  below  twenty-two  and  one-half  cents  per  hour, 
figured  upon. 

Cost  of  cement  varies  with  the  market  and  distance  of  the  work 
from  the  nearest  mill  from  eighty  cents  or  a  dollar  per  barrel  to  two 
or  three  dollars. 

Cost  of  crushed  stone  varies  with  the  locality  and  distance  of 
the  work  from  railroad  or  crushing  plant. 

In  Minneapolis  and  St.  Paul,  from  $1.25  to  $1.75  on  the  work. 
Milwaukee,  $1.25  to  $1.59.  Washed  gravel,  Ohio  river  points, 
$1.00  to  $1.25. 

Cost  of  carting  and  hauling  must  be  investigated  in  each  indi- 
vidual case.  In  many  of  the  smaller  towns  good  concrete  gravel 
can  be  secured  as  low  as  thirty  to  fifty  cents  per  cubic  yard  and  in 


356  COST    OF    REINFORCEMENT 

order  to  give  a  clear  idea  as  to  the  general  questions  of  cost  these 
variables  must  be  carefully  considered  and  investigated  by  the 
bidder  if  figuring  reasonably  close. 

In  securing  this  essential  information  the  conservative  business 
man  will  secure  quotations  in  writing,  especially  when  not  personally 
acquainted  with  the  reliability  of  the  parties  quoting;  then  if  the 
work  is  secured  he  may  at  his  option  hold  the  bidder  to  his  price  or 
seek  redress  by  suit. 

In  giving  the  foregoing  average  values  it  should  be  noted  that 
the  cost  of  placing  varies  eight  to  ten  per  cent  with  the  character 
of  the  reinforcement.  Where  there  are  numerous  beam  boxes  and 
stirrups  and  increased  work  of  puddling  the  concrete,  the  cost 
may  readily  run  five  or  six  per  cent  above  the  average  while  where 
there  is  a  plain  flat  slab  such  as  the  mushroom  system  the  cost  will 
readily  run  five  or  six  per  cent  lower  than  the  average  given. 

15.  Cost  of  Steel.  Medium  Steel,  Open  Heath  or  Bessemer, 
Manufacturers'  Standard  specification  is  at  present  writing  at  $1.20 
base,  Pittsburg. 

The  base  price  is  given  in  all  the  engineering  and  iron  trade 
papers.  All  bars  from  f "  rounds  to  three  inches  are  base.  Smaller 
bars  are  sold  at  base  plus  card  extras. 

The  following  is  the  standard  steel  classification : 

Extra] 

3/4"  to  3" Base 

5/8"  to  11/16" 05 

1/2"  to  9  /16" 10 

7/16" 20 

3/8" 25 

5/16" 30 

1/4"  to  9/32" 35, 

1  to  6"x3/8  to  1"  ...  .Base iFlats  and  heavy 

1  to  6"xl/4  to  5  /16" 20  J     bands. 

The  above  are  full  extras  and  such  sizes  as  we  can  ordinarily 
use  to  advantage  in  concrete  steel  construction. 

In  figuring,  take  base  price  plus  extras  plus  freight  to  destination. 
Freight  rates  to  all  points  in  the  United  States  and  the  Dominion 
of  Canada  are  given  in  compact  form  in  a  book  published  by  the 
American  Steel  &  Wire  Company. 

Cost  of  deformed  bars  rolled  to  standard  specification  at  the 


Rounds  or  square. 


COST    OF    BENDING,    HOOPING,    ETC.  357 

present  writing,  one  dollar  per  ton  above  plain  bars.  Special  rein- 
forcement sold  with  design  from  eight  to  twelve  dollars  per  ton 
additional. 

16.  Cost  of   Bending.     Medium  steel  with  proper  equipment 
rods  for  the  mushroom  system  can  be  bent  cold  for  fifty  to  sixty 
cents  per  ton;  where  high  carbon  steel  is  used  and  rods  are  heated, 
one  and  a  half  to  two  dollars  per  ton  for  bending. 

Beam  rods,  such  as  are  used  in  Turner  beam  system,  can  be 
bent  for  from  two  to  two  and  a  half  dollars  per  ton;  where  more 
complicated  bends  are  employed,  from  two  and  a  half  to  three  and 
a  half  per  ton. 

Cost  of  placing  steel,  including  handling  and  bending,  in  Mush- 
room flat  slab  work  has  run  from  six  to  ten  dollars  per  ton.  With 
a  beam  system  in  beams  spaced  four  to  six  feet  centers  a  cost  of  ten 
to  twelve  dollars  would  be  a  fair  basis  upon  which  to  figure. 

17.  Cost  of  Hooping  for  Columns.     Spirals  made  at  the  shop 
can,  at  the  present  time,  be  furnished  at  a  less  cost  than  they  can 
be  fabricated  in  the  field,  except  in  parts  of  Canada  where  this 
statement  may  not  be  true.     Shop  work  is  better  done  than  field 
work  and  should  be  preferred  other  things  being  equal. 

18.  Cost  of  Centering.     No  item  in  concrete  construction  is 
so  generally  underestimated  as  the  cost  of  false  work  for  reinforced 
concrete.     In  fact,  so  generally  is  this  the  case  that  the  contractor 
inexperienced  in  this  kind  of  work  is  more  than  likely  to  underbid 
those  possessing  both  equipment  and  experience   by  reason  of  under- 
estimating this  item  of  cost. 

Cost  of  the  centering  per  foot  of  floor,  including  columns  and 
beams,  will  vary  anywhere  from  six  to  more  than  twenty  cents  per 
square  foot,  depending  on  the  following  items : 

1.  The  number  of  beam  boxes,  whether  they  frame  into  each 
other  or  into  column  boxes  only. 

2.  The  number  of  columns  for  a  given  floor  area. 

3.  The  number  of  stories  or  floors  that  are  alike. 

4.  The  rapidity  with  which  it  is  desired  to  push  the  work  and 
whether  the  weather  conditions  are  favorable  for  the  prompt  removal 
of  the  forms. 

Where  the  building  has  a  full  concrete  skeleton,  centering  costs 
generally  are  a  third  more  than  where  bearing  walls  are  used. 
Evidently  the  greater  the  number  of  stories  the  more  times  the  lumber 


358  COST    OF    CENTERING 

may  be  moved  up  and  used  over.     Where  the  work  is  to  be  rushed 
rapidly  in  cold  weather  a  larger  amount  of  lumber  is  required. 

The  practical  constructor  is  inclined  to  check  his  estimate  of 
cost  on  the  basis  of  so  much  per  foot  of  floor  for  centering  for  rein- 
forcement and  concrete,  and  estimating  in  this  rough  way  will 
generally  detect  any  error  of  more  than  four  or  five  per  cent  in  an 
elaborate  detailed  estimate. 

Attention  may  be  called  to  the  following  elements  necessary  in 
any  estimate: 

1.  Lumber  required,  nails  and  fastenings. 

2.  Carpenter  labor  of  framing  beam  boxes,  column  boxes,  etc., 
per  thousand  feet. 

3.  Labor  of  setting  plain  slab  forms. 

4.  Labor  of  taking  down  forms  and  moving  up  to  upper  story 
per  thousand  feet  B.  M. 

5.  Waste  of  lumber  and  value  of  old  centering. 

Under  1,  the  amount  of  lumber  required,  it  should  be  observed 
that  the  amount  will  vary  with  the  type  of  design.  In  such  a  type 
as  the  Mushroom  system,  Type  IV,  Fig.  14,  there  must  be  for  the 
sheathing  approximately  one  foot  B.  M.  for  each  flat  foot  of  floor 
area.  For  the  joist,  f  of  a  foot  B.  M.  per  flat  foot  of  floor  area. 
For  ledgers,  one-third  of  a  foot,  B.  M.  for  each  flat  foot  of  floor. 
For  uprights,  f  of  a  foot  B.  M.  for  each  flat  foot  of  floor.  For  col- 
umns, spacing  18'  centers,  from  one-third  to  one-half  foot  B.  M.  for 
each  flat  foot  of  floor.  Total,  about  three  and  one-quarter  or  three 
and  one-half  feet  B.  M.  per  foot  of  floor. 

If  the  work  is  to  be  pushed  rapidly  it  is  necessary  to  figure, 
under  favorable  conditions  for  centering,  not  less  than  two  complete 
floors  of  centering  plus  waste.  If  the  weather  conditions  are  un- 
favorable there  should  be  enough  lumber  for  centering  for  three  to 
four  floors.  On  a  building  having  eight  stories  we  would  ordinarily 
figure  enough  centering  for  three  floors,  plus  waste.  With  the 
flat  slab  system  there  is  no  waste  with  the  joists  as  they  are  simply 
lapped  by  and  the  waste  in  the  boards  would  amount  to  about  two 
per  cent  each  time  they  are  used.  There  will  be  some  waste  in  the 
uprights  if  the  stories  are  of  different  heights,  which  must  be  figured 
in  each  individual  case. 

Where  a  beam  S3rstem  is  used  the  waste  will  be  much  greater  as 
the  loss  from  breakage  and  cutting  the  lumber  to  the  size  of  the  panels 
will  generally  run  the  waste  up  to  ten  to  twenty  per  cent  of  the 


COST    OF    FRAMINd    CENTERING  359 

lumber  in  each  floor,  and  sometimes  much  more  than  this.  Also 
the  surface  contact  is  increased  by  the  area  of  the  sides  of  all  beams 
requiring;  additional  lumber. 

Cost  of  Framing.  Labor  for  framing  beam  boxes,  column 
boxes,  etc.,  will  generally  run  about  twelve  dollars  per  thousand 
feet  B.  M.  Labor  of  placing  plain  slab  forms,  carpenter's  wages, 
being  figured  at  37|c  per  hour,  will  run  about  five  to  six  dollars 
per  thousand  feet.  The  cost  of  taking  down  the  forms  and  moving 
them  up  should  run  about  three  dollars  per  one  thousand  feet  B.  M., 
for  the  flat  slab  type  and  five  to  seven  dollars  per  thousand 
where  there  are  a  large  number  of  beam  boxes,  etc.  Nails  and 
fastenings  are  generally  a  small  item. 

Where  sheet  metal  is  used  for  the  sheathing  the  cost  per  foot 
of  laying  it  and  greasing  it  with  paraffine  is  about  one-third  the  cost 
of  placing  boards,  altho  the  first  cost  of  the  metal  is  considerably 
higher. 

Mr.  L.  C.  Wason,  president  of  the  Alberthaw  Construction 
( Company,  of  Boston,  at  the  fifth  annual  convention  of  the  National 
Association  of  Cement  Users  at  Cleveland,  Ohio,  presented  a  paper 
on  costs  from  which  the  following  table  is  condensed,  giving  the 
cost  of  handling  and  some  very  interesting  costs  of  centering.  It 
would  be  well  for  the  reader  to  look  up  this  paper  which  is  reprinted 
in  part  in  the  Engineering  News,  January  14,  1909,  and  a  number 
of  the  other  engineering  papers. 

The  following  table,  condensed  by  the  Engineering  News,  from 
the  original  paper,  is  given  as  a  fair  indication  of  the  variation  in 
cost  of  different  designs  and  different  conditions.  The  author 
states  that  only  typical  cases  are  given  where  the  items  of  cost 
were  accurately  known.  Enough  are  given  for  a  fair  average  except 
in  the  case  of  long  span  flat  slab  which  appears  to  him  by  comparison 
a  recent  type  of  construction. 

By  reference  to  the  general  averages  on  form  work  in  the  accom- 
panying tables  the  cost  of  forms  per  square  foot  of  surface  contact, 
namely:  Columns,  $0.13;  floors  with  reinforced  concrete  beams, 
$0.116;  flat  floors  without  beams,  $0.111;  short  span  slabs  between 
steel  beams  including  the  fireproofing  on  the  side  of  the  beams, 
$0.05;  walls  exposed  to  view  above  ground,  $0.093;  the  writer  be- 
lieves are  all  higher  in  price  than  usually  believed  to  be  a  fair  cost 
by  most  builders.  It  is  upon  the  success  of  handling  forms  that 
good  results  financially  depend.  In  regard  to  concrete,  labor  is 


360 


ABEBTHAW    CONSTRUCTION    COMPANY    COSTS 


TABLE  1.— SHOWING  COST  OF  FORMS  AND  CONCRETE  ON  VARIOUS  MEMBERS  IN 
REINFORCED-CONCRETE  STRUCTURES 

PLAIN  CONCRETE  COLUMNS 


Forms  per  sq.  ft.                     Concrete  per  cu.  ft. 

Carpen- 

Nails                Con-  Gen- 

Team 

Location                                 ter     Lum- 

and  Total    crete  eral       Ce-  Aggre 

>-  and    Plan 

t  To- 

Labor    ber 

Wire              Labor  Labor  ment   gate  Misc. 

tal 

Office  building,  Portland,  Me... 

$.133 

$.039  $.011 

$.173 

$.064 

$.004 

$.087 

$.084 

$.012 

$.022$.273 

Coal  pocket,  Lawrence,  Miss.  .  . 

.057 

.024 

.001 

.082 

.166 

.003 

.073 

.041 

.008 

.016 

.307 

Mill,  Southbridge,  Mass  

.097 

.082 

.002 

.181 

.073 

.056 

.107 

.035 

.027 

.030 

.328 

Mill,  Attleboro,  Mass  

.093 

.022 

.001 

.116 

.110 

.014 

.062 

.038 

.013 

.034 

.271 

Mill,  Southbridge,  Mass  

.080 

.056 

.001 

.137 

.108 

.048 

.100 

.037 

.013 

.031 

.340 

Coal  pocket,  Hartford  Ct  

.098 

.047!    .002 

.147 

.089 

.043 

.069 

.055 

.017 

.013 

.286 

Garage,  Brookline,  Mass  

.071 

.051 

.002 

.124 

.070 

.028 

.072 

.058 

.041 

.020 

.289 

Warehouse,  Portland,  Me  

.118 

.016 

.001 

.135 

.087 

.027 

.087 

.070 

.039 

.025 

.335 

Textile  mill,  Lawrence,  Miss  .  .  . 

.061 

.013 

.001 

.075 

.095 

.019 

.109 

.027 

.018 

.015 

.283 

Highest  

.133 

.082 

.002 

.181 

.166 

.056 

.109 

.084 

.041 

.034 

.340 

Lowest  

.057 

.013 

.001 

.075 

.064 

.003 

.062 

.027 

.008 

.013 

.271 

Average  of  9  

.082 

.001 

.130 

.096 

.027 

.085 

.049 

.021 

.023 

.301 

REINFORCED-CONCRETE  BEAM  FLOORS 

Highest 

.1651    .1071    .004 
.037]    .027    .001 
070i     045      002 

.275!    .1861   .0351    .1941    .101 
.067    .047    .004    .071    .037 
.  1  1  6       111       020      1  06      063 

.0521    .0551    .470 
.007    .010    .202 
.025|    .024|    -354 

Lowest  
Average  of  18 

FLAT  SLAB  FLOORS 

Highest  

.0781    .0391    .003 

.118 

.146!    .017 

.109!    .084 

.0261   .0391    .374 

Lowest  

.067    .0371    .001 

.106 

.043    .004 

.087    .053 

.012    .010    .252 

Average  of  3 

O71      O38 

OO2 

111 

O97      O09 

O96      070 

019      024 

.315 

REINFORCED-CONCRETE  SLABS  BETWEEN  STEEL  BEAMS 

Highest 

HOI     071 

0031     184 

144 

048 

.208 

.080 

O64 

046 

428 

Lowest  

.028    .012 

.001    .049 

'073 

.005 

.076 

.026    .004 

010 

.272 

Average  of  13  

.06l|    .032 

.002|    .095 

.102 

.019 

.128 

.068|    .024 

!017 

.  359 

BUILDING 

WALLS  ABOVE  GRADE 

Highest  

.1361    .073 

.005!   .176!   .1461    .0521   .105!   .187 

.0771    .055 

.446 

Lowest  

.046    .016 

.001    .079    .042    .004    .034    .043 

.007    .005    .174 

Average  of  17  

.085    .036 

.002    .128    .090    .016)    .073    .076 

.025|    .0191     301 

FOUNDATION  WALLS 

Highest 

1341     048 

0041     1931     213 

037]     203 

116 

Orl7i      O4O 

599 

Lowest  

'.032\    '.009 

.001    ^056    !040 

'.0021    '.038 

'.027\    .0031   .010 

.148 

Average  of  14 

OfiS       033 

nna     i  na     O?R 

01  s     oso 

Ofi2 

.019|   .017 

2B^ 

FOOTINGS  AND  MASS  FOUNDATIONS 

Highest  

.1191    .077 

.0031   .1981   .0811   .0201   .098!   .099 

.013 

.049 

.  275 

Lowest  

.016    .006 

.001    .018    .025    .001    .047    .043 

.003 

.010 

.181 

Average  of  10  057|    .034 

.002]    .093|    .045)    .0071    .07l|    .077 

.007 

.021 

.220 

the  variable  item  which  must  be  carefully  considered.  Any  person 
of  intelligence  can  make  a  careful  estimate  of  the  materials  to  be 
used,  but  note  the  average  prices  of  labor  per  cubic  foot  of  concrete, 
namely:  For  columns,  $0.123;  beam  floors,  $0.131;  flat  floors, 
$0.106;  floors  between  steel  beams,  $0.121;  walls,  $0.106;  foundations, 
$0.091;  and  mass  work  in  connection  with  buildings,  $0.052.  Not 
until  the  last  item  is  a  price  reached  which  according  to  observa- 
tion and  experience  must  be  expected  to  obtain  in  ordinary  building- 
work;  Many  who  have  had  wide  experience  in  handling  large 
quantities  of  concrete  in  mass  have  occasionally  attempted  a  lighter 
type  of  construction  and  have  been  greatly  surprised  at  the  large 
expense  connected  therewith.  Men  with  this  experience  have 
frequently  added  fifty  to  one  hundred  per  cent  to  the  cost  of  mass 
work  and  only  by  doing  so  have  they  felt  that  they  were  sufficiently 
covered  for  light  structural  work. 


ABERTHAW    CONSTRUCTION    COMPANY    COSTS 


361 


Table  II  is  an  exact  copy  of  a  "master  card"  which  gives  the 
complete  financial  history  of  the  job,  when  it  is  finally  completed. 
The  first  column,  which  is  blank,  is  occasionally  used  for  an  estimate 
of  the  first  cost,  the  proposal  including  the  profit  as  well  as  the  esti- 
mated actual  cost.  It  will  be  seen  that  on  some  items,  a  loss  was 
incurred,  as  well  as  a  profit  on  others,  showing  that  it  is  difficult  to 
reach  the  right  price  on  everything,  even  on  work  on  which  a  company 
is  fairly  experienced,  and  also  that  when  slight  changes  are  made 
by  the  owner  or  architect  they  often  entail  heavy  loss  even  though 
the  changes  appear  to  be  extremely  trivial.  Take  the  case  of  the 
external  walls.  The  owners  furnished  the  window  frames  and 
sash,  which  were  all  of  metal.  The  original  design  was  for  a  frame 
with  two  sash,  which  could  easily  be  put  into  a  six-inch  wall. 

TABLE  II.— TYPICAL  "MASTER  CARD." 


Job  No.  747.     Date,  May  24th,  1906.     Mill,  Tappan  Bros.,  Attleboro,  Mass. 

Proposal          Actual  Cost     Per  cu.  ft.        Profit 

Loss          % 

Total  
Excavate 

$35,164.55 
790.00 

$31,330.48 
823.18 

$6  021 

$3,834.071      
$33   18 

11 

Footings  and  Fn  

1,738.00 

1,033.57 

.137 

704.43!      

Per  sq.  ft. 

Exterior  walls  

1,955.00 

2,162.02 

.190 

207  .  02 

Wall  and  Fn.  centers  

1,520.00 

3,630.08 

.125 

2,110.08 

Floors  6|  ins.  thick  

8,883.00 

6,544.16 

.339 

2,338.84 

Roof  5j  ins.  thick  

2,869.00 

1,713.51 

.237 

1,155.49 

Per  lin.  ft. 

Columns,  20  ins.  x  20  ins  

832.00 

676.65 

1.470 

155.35 

Stairs  

883.00 

910.35 

.912 

27^35 

Per  sq.  ft. 

Tool  surface  

469.00 

636.53 

.056 

167.53 

Ornaments  and  cornice  

348.00 

164.33 

183.67 

Ventilators  on  roof  

44.00 

35.64 

8.36! 

Each 

Set  windows  and  door  frames  .  . 

852.00 

929.99 

2.19 

122.01 

Interior  partitions  

1,770.25 

1,656.35 

Per  sq.  ft  . 
.189 

113.90       

Bolts  and  iron  work  

253.00 

257.06 

4.06 

Stair  railing  and  grill  

387.00 

654.00 

267.00 

PerM. 

Screeds  and  settings  

1,086.00 

835.12 

52.17 

250.88 

2-in.  Spr.  plank  and  laying.  .  .  . 

2,839.00 

1,431.69 

33.30 

1,40731 

f-in.  maple  

1,738.00 

1,788.88 

89.44 

50.88 

Motor  shaft 

379  50 

533  19 

98.89 

153.69 

Motor  shaft  found  

98.00 

70.07 

27.93 

Roofing  and  conductors  

1,255.00 

1,026.06 

228.94 

Per  sq.ft. 

Paving  

1,009.00 

747.54 

.094 

361.46 

Retaining  wall  centers,  per  sq.  ft 

.211 

Retaining  wall,  concrete  per  c.  ft 

'  429.00 

316.90 

.175 

l'l2.l6 

Painting  

400.00 

375.00 

25.00 

Steel  footings  and  walls  

300.00 

218.91 

81.09 

Plant,  frt.,  etc  

1,860.00 

2,271.73 

411.73 

Bond  

100.00 

120.00 

20.00 

Extras  

77.80 

67.97 

9.83 

They  later  decided,  for  greater  fire  protection,  to  use  four  sash. 
This  required  an  eight-inch  wall  instead  of  a  six  inch,  and  the  form 
work  on  the  inside  had  to  be  built  inward  and  then  the  space  under 
the  windows  paneled  to  save  material.  To  save  making  a  very 
narrow  panel  at  the  side  of  the  window,  which  would  cost  more  than 
the  concrete  saved,  the  space  was  filled  up  solid  so  that  the  columns 
appear  to  be  wider  than  they  were  actually  figured.  This  slight 


362  SEASON  OF  YEAK,  DEAD  CHARGES,  ETC. 

change,  which  did  not  appear  great  at  the  time,  when  the  job  was 
entirely  complete  showed  that  the  concrete  of  the  walls  showed  an 
actual  loss  instead  of  profit  because  the  form  work  cost  more  than 
twice  what  was  originally  estimated  that  it  should  cost. 

19.  Season  of  Year.     The  season  of  the  year  has  to  be  con- 
sidered in  its  relation  to  the  cost  of  reinforced  concrete  work.     In 
the  summer  season  when  the  concrete  dries  out  rapidly  the  forms 
may  be  removed  every  ten  to  twelve  days,  while  in  the  fall  and  early 
spring  during  frosty  weather  the  water  must  be  heated  or  the  forms 
left  in  place  longer,  requiring  more  lumber  for  centering.     In  the 
winter  when  the  materials  must  be  heated  by  artificial  heat  and 
artificial  heat  used  in  sweating  out  the  concrete,  the  cost  of  work 
will  be  increased  from  ten  to  twelve  per  cent.     Additional  cost  of 
merely  heating  the  water  is  of  course  small.     In  the  chilly  weather 
of  fall  or  spring  good  results  may  be  frequently  obtained  merely 
by  turning  the  exhaust  steam  into  the  water  barrel  and  warming 
the  water  up  so  that  the  concrete  will  set  quickly  notwithstanding 
the  chilly  temperature. 

20.  Dead  Charges.     No  contracting  firm  can  do  business  with- 
out a  considerable  general  expense,  which  must  be  distributed  over 
all  work  executed  by  them.     This  expense  includes  office  expense, 
advertising,  soliciting  work,  estimates  on  not  only  the  work  taken 
but  the  work  which  the  concern  fails  to  secure,  depreciation  of  the 
plant,  freight,  storage  and  equipment,  the  cost  of  keeping  the  organ- 
ization together  in  slack  periods.     This  expense  may  readily  vary 
with  various  concerns  from  five  to  seven  per  cent  of  the  cost  of  the 
work  executed.     In  addition  to  this  dead  expense  and  the  actual 
cost  of  labor  there  must  be  included  the  item  for  liability  insurance 
which  the  contractor  cannot  afford  to  neglect  to  carry.     Frequently 
the  owner  requires  a  surety  bond  for  the  faithful  execution  of  the 
work  and  the  payment  of  bills,  the  cost  of  which  must  be  added  to  the 
incidental  charges  in  the  estimate  of  cost. 

21.  General  Data  on  Cost.     The  architect  is  in  the  habit  of 
figuring  the  building  as  so  much  per  cubic  foot.     For  heavy  ware- 
houses with  the  plainest  kind  of  finish  and  large  size  the  cost  per 
cubic  foot  may  run  as  low  as  six  and  one-half  to  seven  cents  up  to 
ten  and  twelve  cents  for  the  smaller  size  of  buildings  with  office 
fixtures,  plumbing  and  the  like.     No  approximate  cost  per  cubic 
foot  of  any  value  can  be  given  for  office  buildings,  hotels  and  the 
like,  since  this  item  would  vary  greatly  with  the  character  and  differ- 
ence in  the  quality  of  the  finish,  fittings  and  the  like. 


GENERAL    DATA    ON    COST  3()!> 

For  the  concrete  end  of  the  building,  however,  a  rough  approxi- 
mate estimate  can  be  made  very  readily  by  figuring  a  unit  price 
per  square  foot  of  floor  area.  In  a  large  building  of  six  or  seven 
stories  having  a  floor  area  of  twenty  to  thirty  thousand  feet,  panels 
approximately  eighteen  feet  square,  labor  as  outlined,  sand  at 
$1.00  per  yard,  cement,  $1.20,  crushed  stone,  $1.40,  capacity  of  floors 
three  hundred  pounds  per  foot;  rough  slabs,  columns  and  footings 
may  be  erected  at  an  approximate  cost  of  the  contractor  of  about 
forty  cents  per  square  foot  of  floor  area.  Where  the  building  is 
narrow  and  there  are  more  columns  in  proportion  to  the  floor  area, 
on  the  same  basis  fifty  cents  per  square  foot  would  be  a  reasonable 
price. 

Reduction  in  the  floor  load  carried  makes  a  relatively  small 
reduction  ia  the  cost  of  the  construction,  since  the  centering  would 
be  the  same  for  the  light  and  the  heavy  building. 

Where  the  load  is  increased  fifty  per  cent  above  these  require- 
ments the  additional  cost  would  be  increased  over  eight  per  cent. 
While  doubling  the  load  would  not  increase  the  cost  over  about 
ten  or  eleven  per  cent. 

This  is  the  general  type  of  information  the  shrewd  contractor 
carefully  figures  out  for  himself  and  which  enables  him  quickly  and 
accurately  to  check  up  estimates  made  by  his  assistants  or  even  to 
take  work  on  an  approximate  estimate  of  this  kind  without  going 
into  details.  Turner,  when  pressed  for  time  once  took  a  $60,000 
contract  on  a  twenty-minute  estimate  based  on  a  computation 
only  of  the  floor  area  and  a  knowledge  of  the  conditions  covering 
labor  and  cost  of  materials. 

Where  there  are  plain  reinforced  floors  resting  on  walls  and  the 
panels  are  of  large  size  such  as  in  court  house  work  and  many  other 
public  buildings  and  where  gravel  can  be  cheaply  obtained  the  cost 
per  foot  of  floor  may  run  as  low  as  30  cents  per  square  foot.  In 
other  localities  forty  cents  per  foot  under  less  favorable  conditions 
would  be  a  reasonable  figure. 


364 


CHAPTER  XI 

1.  Fireproof  Properties  of  Concrete  and  the  Protection  of  Steel 
from  Heat.  The  value  of  concrete  as  a  fireproof  material  has  been 
pretty  well  demonstrated  in  a  large  number  of  severe  conflagrations 
and  also  in  many  fire  tests  by  the  building  departments  of  various 
cities.  In  fact,  it  may  be  stated  that  concrete  ranks  as  the  best 
fireproof  building  material  and  it  is  to  this  quality  together  with 
its  low  cost  that  the  enormous  increase  in  its  use  is  due. 

Intense  heat  injures  the  surface  of  the  concrete,  but  it  is  so 
good  a  non-conductor  that  if  sufficiently  thick  it  provides  ample 
protection  for  the  steel  reinforcement  and  the  interior  of  the  mass 
is  unaffected  even  in  unusually  severe  fires. 

For  efficient  fire  protection  in  slabs  under  ordinary  conditions 
with  one-way  reinforcement  the  lower  surface  of  the  steel  rods 
should  be  f"  above  the  bottom  of  the  slab.  With  two-way  rein- 
forcement this  may  be  reduced  to  \" ,  for  in  case  one  layer  of  rods 
should  become  overheated  the  upper  layer  is  still  amply  protected. 

Structural  beams,  girders  and  columns  should  have  at  least 
2J"  of  good  concrete  for  efficient  protection.  In  beams  having 
large  rods  the  thickness  of  the  concrete  coating  outside  of  the  rods 
should  never  be  less  than  1J"  nor  less  than  the  diameter  of  the  largest 
rod  used  in  the  beam.  In  columns  the  shell  outside  the  reinforce- 
ment should  be  considered  as  fire  protection  and  no  dependence 
placed  upon  it  in  figuring  the  strength  of  the  section,  in  carrying 
the  working  load. 

These  limitations  are  sufficient  for  ordinary  purposes.  Where, 
for  example,  a  factory  building  is  to  be  erected  in  which  there  will 
be  scarcely  any  inflammable  materials  to  be  stored,  it  is  a  waste 
of  money  to  provide  a  thick  concrete  protection  to  resist  possible  fire. 
On  the  other  hand,  where  the  building  is  to  be  used  for  storage  of 
material  capable  of  creating  not  only  a  hot  fire  but  an  intense  heat 
of  long  duration,  special  provision  may  be  made  by  using  an  excessive 
thickness  of  concrete  for  fire  protection  tho  in  such  a  situation  a 
sprinkler  system  would  be  preferable. 


FIRE    RESISTANCE    OF    CONCRETE  3t)5 

A  most  severe  practical  test  occurred  in  a  fire  at  the  Pacific 
Coast  Borax  Refinery  at  Boyanne,  N.  J.  This  building  was  a  four- 
story  factory  built  entirely  of  reinforced  concrete  except  the  roof. 
The  contents  of  the  building,  the  roof  and  interior  wood  trim  were 
destroyed,  but  the  walls  and  floors  remained  intact  except  where 
an  eighteen  ton  tank  fell  thru  the  roof  and  cracked  some  of  the 
floor  beams.  The  heat  was  so  intense  that  brass  and  iron  castings 
were  melted  to  junk.  A  small  annex  built  of  structural  steel  frame 
was  completely  wrecked  and  the  metal  bent  and  twisted  into  a  tangled 
mass. 

In  general,  the  fire  resistance  of  Portland  cement  concrete  is 
governed  or  affected  by  the  character  of  the  aggregate  and  the 
amount  of  cement  in  the  mortar. 

First,  we  may  state  it  as  a  general  rule  that  the  richer  the  mortar 
or  the  greater  the  amount  of  cement  used  the  greater  the  fire  resist- 
ing properties  of  the  concrete.  Rich  mortar  makes  a  stronger 
concrete  better  able  to  resist  severe  temperature  stresses  while  the 
high  proportion  of  cement  when  dehydrated  on  the  exposed  surface 
makes  a  very  perfect  non-conducting  material,  preventing  the 
uninjured  parts  from  further  or  rapidly  progressive  injury. 

Second,  as  regards  the  aggregate,  the  smaller  the  stone  the  better 
the  fire  resisting  qualities. 

Trap  rock  will  make  a  concrete  offering  greater  resistance  to 
extreme  heat  than  limestone  or  granite. 

In  a  series  of  experiments  to  determine  the  effect  of  very  high 
temperatures  on  concrete,  with  the  acetylene  oxygen  blow-pipe, 
interesting  results  were  secured.  The  heat  of  this  flame  is  approx- 
imately 6500°  Fahr.  Applied  to  concrete  paving  block  2  inches 
thick  and  five  years  old,  made  of  sand  and  gravel,  the  heat  under 
the  flame  was  sufficient  to  melt  the  silica  sand  and  form  a  little 
puddle  of  glass.  Pebbles  of  feldspar  or  granite  under  this  intense 
heat  popped,  but  the  little  puddle  of  glass  once  formed  did  not 
seem  to  increase  under  the  continued  application  of  the  flame  and 
hardened  up  as  soon  as  the  flame  was  removed. 

This  series  of  experiments  was  continued  by  using  a  concrete 
of  silica  sand  mixed  with  a  higher  percentage  of  brine  and  it  was 
found  with  such  concrete  possible  to  glaze  the  surface  in  this  manner, 
while  the  concrete  back  of  the  glazing  did  not  seem  to  be  materially 
injured  in  point  of  strength.  Whether  a  concrete  block  of  selected 
materials  can  be  glazed  in  this  manner  uniformly  is,  of  course,  open 


366  FIRE    TESTS 

to  question,  but  any  effort  to  cut  concrete  by  intense  heat  as  steel 
is  cut,  was  proved  by  this  work  to  be  impracticable. 

2.  Fire   Tests.     Building   departments   sometimes   require   fire 
tests  of  the  finished  construction.     A  test  required  in  the  Railway 
Exchange  Building,  Denver,  Colo.,  is  of  interest  from  the  fact  that 
two  tests  were  made,  one  on  thoroly  cured  concrete  and  the  other 
on  concrete  not  well  cured  nor  dried  out.     The  first  test  was  a  fire 
consisting  of  a  cord  of  pine  wood  split  in  faggots  about  two  inches 
square  and  soaked  with  oil,  applied  to  the  under  side  of  the  first 
floor  slab.     The  fire  gave  a  very  intense  heat  and  as  it  dried  down 
a  fire  hose  was  turned  on  the  white  hot  surface.     The  damage  to 
the  slab  consisted  in  the  spalling  of  an  area  about  two  feet  in  diameter 
to  a  depth  of  about  one  and  one-half  inches.     Spalling  was  accom- 
panied by  reports  described  as  being  as  sharp  as  pistol  shots.      The 
cause  of  this  spalling  was  at  first  somewhat  puzzling.     An  examina- 
tion of  the  aggregate  showed  it  to  be  a  good  hard  sandstone  which 
had  been  at  some  period  metamorphosed  by  heat.     There  were, 
however,  numerous  porous  veins  running  thru  the  stone  and  it  seems 
that  these  veins  having  absorbed  considerable  water  in  mixing  the 
concrete  which  had  not  been  dried  out  in  the  curing,  offered  an 
opportunity  for  the  generation  of  steam  in  the  small  cavities  under 
the  heat  of  the  fire  resulting  in  the  bursting  of  the  stone  with  a 
sharp  report.     The  fractures  noted  were  clean  cut.   as  would   be 
expected  from  such  a  cause. 

Allowing  the  slab  to  thoroly  dry  out  for  an  additional  period  of 
five  weeks,  a  second  test  was  made,  similar  to  the  first,  with  abso- 
lutely no  spalling  and  no  apparent  injury  to  the  slab.  Cut  showing 
this  test  is  shown  in  Fig.  93. 

3.  The  Theory  of  Fire  Protection.     The  theory  of  fire  protec- 
tion is  given  by  Mr.  Newberry  as  follows: 

"Two  principal  sources  from  which  cement  concrete  derives 
its  capacity  to  resist  fire  and  prevent  transference  of  the  heat  to  the 
steel  are  its  combined  water  and  porosity.  Portland  cement  takes 
up  in  hardening  a  variable  amount  of  water,  depending  on  surround- 
ing conditions.  In  a  dense  briquette  of  neat  cement  the  combined 
water  may  reach  twelve  per  cent.  A  mixture  of  cement  with  three 
parts  sand  will  take  up  water  to  the  amount  of  about  eighteen  pel- 
cent  of  the  cement  contained.  This  water  is  chemically  combined, 
and  not  given  off  at  the  boiling  point.  On  heating,  a  part  of  the 
water  goes  off  at  about  five  hundred  degrees  Fahr.,  but  the  dehydra- 
tion is  not  complete  until  nine  hundred  degrees  Fahr.,  is  reached. 


FIRE    TEST    IN    THE    RAILWAY    EXCHANGE    BUILDING,    DENVER,    COLO.          367 


Fisher  Bros.,  Architects,  Denver,  Colo. 

Martin  Carroll,  Contractor,  Kansas  City,  Mo. 

Fig.  93. 

This  vaporization  of  water  absorbs  heat  and  keeps  the  mass  for  a 
long  time  at  a  comparatively  low  temperature.  A  steel  beam  or 
column  embedded  in  concrete  is  thus  cooled  by  the  volatilization 
of  water  in  the  surrounding  cement.  The  principle  is  the  same  as 
in  the  use  of  crystallized  alum  in  the  casings  of  fireproof  safes; 
natural  hydraulic  cement  is  largely  used  in  safes  for  the  same 
purpose. 

The  porosity  of  concrete  also  offers  great  resistance  to  the  passage 
of  heat.  Air  is  a  poor  conductor,  and  it  is  well  known  that  an  air 
space  is  a  most  efficient  protection  against  conduction.  Porous 
substances,  such  as  asbestos,  mineral  wool,  etc.,  are  always  used  as 
heat  insulating  material.  For  the  same  reason  cinder  concrete, 
being  highly  porous,  is  a  much  better  non-conductor  than  a  dense 
concrete  made  of  sand  and  gravel  or  stone,  and  has  the  added  advant- 
age of  lightness.  In  a  fire  the  outside  of  the  concrete  may  reach 
a  high  temperature  but  the  heat  only  slowly  and  imperfectly  pen- 
etrates the  mass,  and  reaches  the  steel  so  gradually  that  it  is  carried 
off  by  the  metal  as  fast  as  it  is  supplied." 


3(58  STONE  AND  CINDER  CONCRETE  COMPARED 

In  regard  to  cinder  concrete  it  may  be  added,  first,  that  it 
is  not  a  desirable  material  to  use  from  the  standpoint  of  strength. 
Second,  that  as  usually  employed,  insufficient  cement  is  used  to  make 
a  good  fire  resisting  material.  Thus  Prof.  Norton  compares  the 
action  of  stone  and  cinder  concrete  in  the  Baltimore  fire  as  follows: 

"Little  difference  in  the  action  of  the  fire  on  stone  and  cinder 
concrete  could  be  noted  and  as  I  have  earlier  pointed  out  the  burning 
of  bits  of  coal  in  poor  cinder  concrete  is  evenly  balanced  by  the 
splitting  of  stone  in  the  stone  concrete.  I  have  never  been  able  to 
see  that  in  the  long  run  either  stood  fire  better  or  worse  than  the  other. 
However,  owing  to  its  density,  the  stone  concrete  takes  longer  to 
heat  through." 

Perhaps  if  the  relative  proportion  of  cement  were  the  same  in 
each,  the  cinder  concrete,  if  the  cinders  are  real  clinker,  would 
prove  the  better  fire  resisting  material  as  Mr.  Newberry  assumes. 
This  point  cannot  be  too  much  emphasized. 

A  concrete  must  be  rich  in  cement  to  make  a  first  class  fireproof 
material  and  for  this  reason  alone  a  leaner  mixture  than  1:2:4 
should  not  be  allowed  in  an  important  building. 

Thus  far  our  attention  has  been  primarily  directed  to  the  fire- 
proof qualities  of  concrete  as  such.  In  considering  the  fire  resisting 
properties  of  the  composite  material  known  as  concrete  steel  or 
reinforced  concrete,  the  effect  of  the  unequal  heating  of  different 
parts  of  the  construction  must  be  considered.  It  has  been  pre- 
viously noted  that  the  coefficient  of  expansion  of  steel  and  concrete 
are  practically  identical.  Their  coefficients  of  heat  capacity  and 
conductivity,  however,  differ  widely  and  for  this  reason  the  dis- 
tribution of  the  metal  in  the  form  of  small  bars  rather  than  in  large 
units  will  give  a  more  satisfactory  result  from  the  fireproof  standpoint . 

4.  Terra  Cotta  and  Tile  Compared  with  Concrete.  The  dif- 
ficulty with  the  combination  of  tile  or  terra  cotta  and  structural 
steel  as  a  fire  resting  material  lies  largely  in  the  fact  that  the 
coefficient  of  expansion  of  the  two  materials  is  different. 

This  is  well  illustrated  in  Fig.  94  showing  the  effect  of  heat  in 
breaking  and  cracking  tile  between  steel  beams  after  exposure  to  a 
severe  fire. 

Professor  Norton,  in  his  report  on  the  Baltimore  fire  to  the  Insur- 
ance Engineering  Experiment  Station,  states: 

"Where  concrete  floor  arches  and  concrete  steel  construction 
received  the  full  force  of  the  fire  it  appears  to  have  stood  well, 
distinctly  better  than  the  terra  cotta.  The  reasons  I  believe  are 


CONCRETE    AND    TIL.K    COMPAKKD 


Fig.  94.     Effect  of  Fire  on  Tile  Construction. 

these:  first,  because  the  concrete  and  steel  expanded  at  sensibly 
the  same  rate,  and  hence  when  heated  do  not  subject  one  another 
to  stress,  but  terra  cotta  usually  expands  about  twice  as  fast  with 
increase  in  temperature  as  steel,  and  hence  the  partition  and  floor 
arches  soon  become  too  large  to  be  contained  by  the  steel  members 
which  under  ordinary  temperature  properly  enclose  them.  Under 
this  condition  the  partition  must  buckle  and  the  segmental  arches 
must  lift  and  break  the  bonds,  crushing  at  the  same  time  the  lower 
surface  member  of  the  tiles. 

"When  brick  or  terra  cotta  are  heated  no  chemical  action  occurs, 
but  when  concrete  is  carried  up  to  about  1,000  degrees  Fahr.,  its 
surface  becomes  decomposed,  dehydration  occurs,  and  water  is 
driven  off.  This  process  takes  a  relatively  large  amount  of  heat. 
It  would  take  about  as  much  heat  to  drive  the  water  out  of  this 
outer  quarter  inch  of  the  concrete  partition  as  it  would  to  raise  that 
quarter  inch  to  1,000  degrees  Fahr.  Now,  a  second  action  begins. 
After  dehydration  the  concrete  is  much  improved  as  a  non-conductor 
and  yet  thru  this  layer  of  non-conducting  material  must  pass  all  the 
heat  to  dehydrate  and  raise  the  temperature  of  the  layers  below, 
a  process  which  cannot  proceed  with  great  speed." 


370  REPORT    ON    BALTIMORE    FIRE 

In  the  composite  material  of  concrete  and  steel  in  the  form  of  a 
continuous  concrete  monolith  there  are  severe  temperature  stresses 
set  up  by  the  unequal  heating  of  different  parts  of  a  floor  during  a 
fire  and  the  manner  in  which  the  material  will  withstand  these 
stresses  will  depend  in  a  large  measure  on  how  thoroly  the  steel  is 
disseminated  thru  the  concrete  to  enable  it  to  take  up  the  tensile 
stresses  induced  by  this  unequal  expansion  in  the  various  parts  caused 
by  the  unequal  heating,  hence  that  type  of  construction  which  is 
reinforced  practically  in  all  directions  is  best  calculated  to  with- 
stand the  severe  stresses  so  produced.  Further,  since  the  concrete 
is  injured  or  disintegrated  on  its  surface  the  smaller  the  surface 
exposed  the  less  will  be  the  damage,  and  the  fewer  irregularities  in 
the  form  of  the  construction,  the  less  it  will  be  injured. 

Looking  at  the  question  from  this  standpoint  then,  the  flat  slab 
type  of  construction  would  rank  first  from  the  fireproof  standpoint 
and  Type  III  second.  In  other  words,  the  natural  concrete  types 
which  are  in  no  wise  imitations  of  older  types  of  construction  are 
far  better  adapted  to  resist  the  severe  conditions  of  a  conflagration 
than  those  types  which  are  merely  imitations  of  older  forms  of 
construction. 

In  reporting  to  the  Chief  of  Engineers,  U.  S.  A.,  regarding  one 
of  the  reinforced  concrete  buildings  which  passed  through  the  Balti- 
more fire,  Capt.  Sewell  writes:* 

"It  was  surrounded  by  non-fireproof  buildings,  and  was  subjected 
to  an  extremely  severe  test,  probably  involving  as  high  temperature 
as  any  that  existed  anywhere.  The  concrete  was  made  with  broken 
granite  as  an  aggregate.  The  arches  of  the  roof  and  the  ceiling 
of  the  upper  story  were  cracked  along  the  crown,  but  in  my  judg- 
ment very  slight  repairs  would  have  restored  any  strength  lost  here. 
Cutting  out  a  small  section —  say  an  inch  wide — and  caulking  it 
full  of  good  strong  cement  mortar  would  have  sufficed.  The  exposed 
corners  of  columns  and  girders  were  cracked  and  spalled,  showing 
a  tendency  to  round  off  to  a  curve  of  about  three  inches  radius.  In 
the  upper  stories,  where  the  heat  was  intense,  the  concrete  was 
calcined  to  a  depth  of  from  J  to  f  inch,  but  it  showed  no  tendency 
to  spall,  except  at  exposed  corners.  On  wide,  flat  surfaces,  the 
calcined  material  was  not  more  than  J  inch  thick,  and  showed  no 
disposition  to  come  off.  In  the  lower  stories,  the  concrete  was 
absolutely  unimpaired,  tho  the  contents  of  the  building  were  all 


*Eng.  News,  March  24,  1904. 


LOAD    CAPACITY    AFTER    EXPOSURE    TO    FIRE  371 

burned  out.  In  my  judgment,  the  entire  concrete  structure  could 
have  been  repaired  for  not  over  twenty  to  twenty-five  per  cent  of 
its  original  cost.  On  March  10th,  I  witnessed  a  loading  test  of  this 
structure.  One  bay  of  the  second  floor,  with  a  beam  in  the  center, 
was  loaded  with  hearly  three  hundred  pounds  per  square  foot  super- 
imposed, without  a  sign  of  distress,  and  with  a  deflection  not  exceed- 
ing |  inch.  The  floor  was  designed  for  a  total  working  load  of  150 
pounds  per  square  foot.  The  sections  next  to  the  front  and  rear 
walls  were  cantilevers,  and  one  of  these  was  loaded  with  150  pounds 
per  square  foot,  superimposed,  without  any  sign  of  distress,  or  undue 
deflection." 

In  concluding  the  subject  of  the  fireproof  qualities  of  concrete 
it  may  be  well  to  call  attention  to  the  stock  argument  of  the  burned 
clay  advocate. 

A  small  specimen  of  burned  clay  or  terra  cotta  if  subjected  to 
a  temperature  of  2,000  degrees  and  then  immersed  in  water  will 
remain  undamaged. 

A  small  sample  of  concrete  subjected  to  similar  treatment  will 
be  totally  disintegrated.  Hence  the  burned  clay  advocate  argues 
that  concrete  is  not  a  suitable  fireproof  material. 

The  fallacy  in  this  plausible  argument  as  has  been  pointed  out 
in  an  excellent  editorial  in  the  Engineering  News  lies  in  the  fact 
that  the  conditions  in  a  building  during  a  fire  and  in  the  furnace 
are  radically  different. 

In  a  fire  in  a  building  the  concrete  is  not  exposed  to  heat  on  all 
sides,  nor  is  it  exposed  continuously  for  any  long  time  to  very  high 
temperatures.  The  greatest  heat  is  generally  near  the  ceiling  when 
the  surface,  as  noted  in  Capt.  Sewall's  report,  may  be  dehydrated 
slightly  and  protect  the  material  back  of  the  injured  portion.  The 
net  result  is  that  less  damage  results  than  to  the  terra  cotta  or 
hollow  tile,  since  the  latter  does  not  expand  in  unison  with  the 
supporting  steel  frame,  and  is  crushed  and  broken  by  the  severe 
temperature  stress  resulting  from  this  cause. 

Combination  structures  of  hollow  tile  and  concrete  are  open  to 
the  same  criticism  from  the  fireproof  standpoint,  namely,  the  com- 
bination of  two  elements  in  a  composite  structure  having  radically 
different  coefficients  of  expansion.  Evidently  the  expectation  that 
the  combination  will,  under  severe  conditions,  prove  satisfactory 
cannot  be  realized. 

5.     Rates  of  Insurance  on  Concrete  Buildings   and  Contents. 

Boards  of  fire  underwriters  representing  the  older  line  companies, 


372  INSURANCE    RATES 

have  been  somewhal  slow  in  recognizing  concrete  as  a  fireproof 
material  and  it  seems  to  1he  concrete  constructor  frequently  that 
they  do  not  recognize  the  great  differences  that  exist  in  this  material 
as  dependent  on  the  character  of  the  mixture  and  dissemination  of 
the  metal  reinforcement. 

The  position  that  some  of  these  boards  have  taken  in  rating  the 
mill  building  with  a  sprinkler  system  lower  than  a  concrete  building 
without  a  sprinkler  is  a  position  hard  to  explain  except  that 
possibly  members  of  these  boards  are  financially  interested  in 
sprinkler  system  equipments. 

On  the  other  hand,  the  mutual  companies  appear  to  have  been 
more  progressive  and  are  writing  policies  at  rates  that  appeal  to 
the  constructor  as  far  more  consistent  and  rational. 

Comparing  the  lowest  rate  which  has  come  to  the  writer's  atten- 
tion for  a  timber  building,  mill  construction,  used  for  mercantile 
purposes,  equipped  with  sprinkler  system,  A.  D.  T.  watchman 
service,  etc.,  with  the  lowest  rate  which  has  come  under  his  notice 
for  a  reinforced  concrete  building  similarly  equipped  with  a  sprinkler 
system,  the  rate  for  the  concrete  building  was  less  than  one-half 
that  for  the  timber  building,  being  a  six-cent  rate  for  the  concrete 
structure  against  a  twelve  and  one-half-cent  rate  for  the  timber 
building.  The  advantages  from  the  fireproof  standpoint  may  be 
stated  as  follows: 

(1)  A  well  designed  reinforced  concrete  building  offers  security 
against  disastrous  fire  and  total  loss  of  structure. 

(2)  It  reduces  the  danger  of  damage  to  the  contents  by  pre- 
venting the  spread  of  fire  from  floor  to  floor. 

(3)  It  prevents  damage  to  the  contents  by  water  from  story  to 
story. 

(4)  It  renders  sprinklers  unnecessary  in  buildings  whose  con- 
tents are  not  especially  inflammable. 

(5)  It  reduces  the  danger  of  panic   and  loss  of  life  incident 
thereto  among  employes  or  occupants  of  the  building. 

Evidently  in  order  to  prevent  the  spread  of  fire  from  floor  to 
floor,  the  floors  should  be  continuous,  or  have  openings  properly 
protected  by  automatic  shutters  or  doors.  Thus,  if  we  are  to  protect 
the  goods  or  contents  on  the  floors  above  from  fire  below,  it  is 
necessary  to  have  the  elevator  shafts  protected  by  automatic  fire 
doors  and  stairways  cut  off  in  a  similar  manner.  This  can  be  done 
at  a  comparatively  small  expense. 


INSURANCE    KATES 


373 


Protection  from  exterior  exposure  may  be  readily  made  by 
the  employment  of  wire  glass,  metal  frames  and  the  like,  in  place  .of 
wood  frames  and  ordinary  glass  windows. 

A  good  concrete  floor  is  practically  waterproof  and  a  slight 
pitch  with  suitable  scuppers  would  practically  eliminate  water 
loss  in  floors  below  from  flooding  a  floor  in  which  fire  has  broken 
out  in  the  contents  or  goods  stored  thereon. 

In  the  ordinary  factory  or  mercantile  building  with  wood  floors, 
loss  from  water  is  frequently  greater  than  the  loss  by  actual  fire 
where  an  incipient  blaze  has  been  extinguished. 

In  the  concrete  building,  on  the  other  hand,  each  floor  becomes 
almost  a  waterproof  roof.  Frequently  a  tenant  moves  into  the  lower 
stories  of  a  concrete  building  before  the  upper  portion  is  complete, 
the  floors  above  acting  as  a  roof. 

According  to  Mr.  Kunhardt,  vice-president  of  the  Boston 
Manufacturers  Mutual  Fire  Insurance  Company,  these  mutual 
companies  take  a  business-like  stand  regarding  the  extent  of  fire 
protection  required  in  each  individual  case.  While  the  value  of 
the  automatic  sprinkler  is  recognized  and  the  general  rule  specifies 
its  installation  the  Factory  Mutual  Companies  do  not  require  it 
in  the  concrete  building  except  where  there  is  sufficient  inflammable 
material  in  the  contents  to  furnish  fuel  for  a  fire. 

An  essential  feature  in  good  factory  construction  includes  not 
only  consideration  of  the  building  but  protection  adequate  to  its 
needs  only.  The  extent  to  which  the  above  is  faithfully  carried 
out  will  eventually  be  the  determining  feature  in  the  cost  of  insurance. 

Mr.  Kunhardt  gives  the  following  table : 


i    Brick  Mill       Wood  Mill      J      If 
All  Concrete    Construction    Construction     ^  » ^  S 
or  or 


Open  Joists  |  Open  Joists 


Bldg 


.  ICont'si 


!  Bldg. 


Conts'i  Bldg.   Cont's:      § 

'.     ~C  3  O  ." 


General  Storehouse 
Wool  Storehouse .  . 
Office  Building.  .  .  . 
Cotton  Factory .  .  . 

Tannery 

Shoe  Factory 

WollenMill 

Machine  Shop .... 
General  Merchandise  Bldg 


20c 

45c 

60c 

lOOc 

lOOc 

12oc 

25c 

20c 

35c 

40c 

60c 

75c 

lOOc 

25c 

15c 

30c 

35c 

50c 

lOOc 

125c 

25c 

40c 

lOOc 

lOOc 

200c 

200c 

300c 

50c 

20c 

40c 

75c 

lOOc 

lOOc 

lOOc 

25c 

25c 

80c 

75c 

lOOc 

I50e 

200c 

50c 

30c 

80c 

75c 

lOOc 

160c 

200c 

oOc 

loc 

25c 

oOc 

50c 

lOOc 

lOOc 

25c 

ildg.  .   35c 

75c 

50c 

lOOc 

lOOc 

150c 

25c 

374  FIRE    AND    WATER    LOSS 

These  costs  are  based  on  the  absence  of  automatic  sprinklers  and 
other  private  fire  protective  appliances  of  the  usual  completely 
equipped  building.  They  are  not  schedule  rates,  but  may  be  an 
approximation  to  actual  costs  under  favorable  conditions  based  on 
examples  in  various  parts  of  the  country. 

As  illustrating  the  value  of  fire  protection,  Mr.  Kunhardt  states, 
that  in  the  Boston  Manufacturers'  Mutual  Companies,  the  average 
cost  of  insurance  on  the  better  class  of  protected  factories  has  now 
for  some  years  averaged,  excluding  interest,  less  than  seven  cents 
on  each  hundred  dollars  of  risk  taken,  and  on  first  class  warehouses 
connected  with  them,  one-half  of  this  amount.  These  figures  can 
be  compared  with  the  table  as  illustrating  the  gain  by  the  installation 
of  proper  safeguards  for  preventing  and  extinguishing  fire. 

In  these  same  protected  factories  and  warehouses  the  actual 
fire  and  water  loss  is  less  than  four  cents  on  each  $100  of  insurance 
and  he  regards  it  possible  to  reduce  this  loss  materially,  practically 
along  the  lines  above  outlined. 

Where  sprinkler  systems  are  installed  in  concrete  buildings, 
and  in  particular  where  these  buildings  are  of  the  flat  slab  type 
which  does  not  interfere  with  the  most  perfect  operation  of  the 
sprinkler,  rates  quoted  as  low  as  10  or  12  cents  for  the  building 
and  15  cents  for  the  contents  are  not  uncommon  at  the  present 
time  (1914),  providing  that  80  percent  of  the  insurable  value  of 
the  building  and  contents  is  covered  and  further  that  the  policy  is 
written  for  the  term  of  five  years. 


375 


CHAPTER  XII 

1.  Protection  of  Steel  and  Iron  from  Corrosion  by  Portland 
Cement.  Deterioration  of  steel  by  corrosion  or  rusting  is  one  of 
the  difficult  problems  in  nearly  all  structures  intended  to  be  per- 
manent. 

Paint  of  linseed  oil  combined  with  some  pigment  is  ordinarily 
used  for  the  protection  of  structural  steel  and  its  efficiency  depends 
on  the  complete  removal  of  rust  before  painting.  Further,  this 
coat  of  paint  must  be  renewed  at  frequent  intervals. 

Fortunately  in  concrete  steel  construction  we  have  in  the  cement 
itself  the  most  perfect  protective  coating  known  for  iron  and  steel. 
If  bars  that  are  somewhat  rusty  be  placed  in  wet  concrete  and  re- 
moved after  one  week  they  will  be  found  to  be  perfectly  clean,  the 
rust  having  been  chemically  destroyed  by  the  cement. 

The  bond  between  cement  and  steel  is  formed  better  with 
bars  that  are  somewhat  rusty  when  placed  in  the  concrete  than 
with  bars  new  from  the  mill.  The  reason  seems  to  be  that  a  small 
amount  of  rusting  removes  the  black  mill  scale  and  allows  the 
cement  to  come  in  contact  with  the  solid  bar.  Paint,  oil  or  grease 
tend  to  weaken  the  bond  of  the  concrete. 

Bars,  removed  from  cement  after  over  twenty  years'  exposure 
of  the  specimens  to  the  elements,  have  been  found  bright  and  as 
good  as  when  first  placed  in  the  work. 

This  protection  however  is  dependent  entirely  oh  the  thoro 
covering  of  the  steel  by  the  wet  concrete  and  hence  the  importance 
of  using  a  plastic  mixture,  or  one  that  will  flow  slowly  and  thoroly 
surround  the  steel,  and  require  only  puddling  rather  than  tamping 
to  secure  substantial  work. 

It  may  be  noted  incidentally  also  that  exactly  the  same  kind 
of  mixture  is  essential  if  we  are  to  secure  smooth  work,  neat  in  ap- 
pearance, that  is,  work  without  ragged  patches,  and  showing  no 
rough  stone  that  expose  voids  not  filled  with  mortar. 

A.  B.  Newberry  states  the  chemical  theory  of  protection  of  iron 
embedded  in  concrete  as  follows: 


37(5  PERMANENCE    OF    CONCRETE    CONSTRUCTION 

''The  rusting  of  iron  consists  of  oxidation  of  the  metal  to  the 
condition  of  hydrated  oxide.  It  does  not  take  place  at  ordinary 
temperatures,  in  dry  air  or  in  moist  air  free  from  carbonic  oxide. 
The  combined  action  of  moisture  and  carbonic  acid  are  necessary. 
Ferrous  carbonate  is  first  formed;  this  is  at  once  oxidized  to  ferric 
oxide  and  the  liberated  carbon  dioxide  acts  on  a  fresh  portion  of 
metal.  Once  started  the  corrosion  proceeds  rapidly,  perhaps  on 
account  of  the  galvanic  action  between  the  oxide  and  the  metal. 
Water  holding  carbonic  acid  in  solution,  if  free  from  oxygen,  soon 
acts  as  an  acid  and  rapidly  attacks  iron.  In  lime  water  or  soda 
solution  the  metal  remains  bright.  The  action  of  cement  in  pre- 
venting rust  is  now  apparent.  Portland  cement  contains  about 
sixty-three  percent  lime.  By  the  action  of  water  it  is  converted 
into  a  crystalline  mass  of  hydrated  calcium  silicate  and  calcium 
hydrate.  In  the  hardening  it  rapidly  absorbs  carbonic  acid  and 
becomes  coated  on  the  surface  with  a  film  of  carbonate.  Cement, 
mortar  thus  acts  as  an  efficient  protector  of  iron  and  captures  and 
imprisons  every  carbonic  acid  molecule  that  threatens  to  attack 
the  metal.  The  action  is,  therefore,  not  due  to  the  exclusion  of  the 
air,  and  even  tho  the  concrete  be  porous,  and  not  in  contact  with 
the  metal  at  all  points,  it  will  still  filter  out  and  neutralize  the  acid 
and  prevent  its  corrosive  effect." 

2.  Permanence  of  Concrete  Construction  when  made  with 
Proper  Materials.  The  best  grade  of  Portland  concrete  made  with 
the  first  class  cement  selected  aggregate,  properly  mixed  and  cured 
is  indeed  a  most  permanent  material,  fully  justifying  all  that  can  be 
said  in  its  favor.  It  will  withstand  the  action  of  the  elements  as 
well  as  granite  and  quartzite,  and  will  withstand  the  heat  of  fire 
better  than  granite,  while  in  small  samples  is  not  equal  to  the  granite 
in  point  of  strength,  in  large  masses  it  may  be  said  that  it  may 
be  depended  upon  with  great  certainty  because  there  are.  no  seams, 
flaws  or  planes  of  weakness  such  as  are  liable  to  be  found  in  masses 
of  natural  stone. 

A  good  concrete  increases  in  strength  with  age  and  grows  harder 
and  stronger  as  time  continues.  The  increase  in  strength  is  rapid 
for  the  first  three  months  and  continues  at  a  gradually  decreasing 
rate  for  the  next  six  or  eight  months,  and  then  very  slowly  as  time 
goes  on,  perhaps  thru  a  period  of  twenty-five  years  or  more. 

Where  the  concrete,  however,  is  not  made  from  suitable  aggre- 
gates, is  not  properly  mixed  and  cured,  it  is  by  no  means  a  permanent 
material  when  exposed  to  the  action  of  the  elements.  Concrete 


CONCRETE    MADE    WITH    IMPROPER    MATERIALS  377 

having  for  an  aggregate  a  soft  stone,  such  as  some  of  the  oolitic 
limestones,  shale  or  one  which  is  made  with  sand  which  is  fine  and 
containing  considerable  clay  will  inevitably  be  affected  materially 
by  frost  in  the  severe  climate  of  the  north. 

Concrete  Made  with  Improper  Materials.  In  building  work 
concrete  is  under  cover  and  in  the  main  protected  from  the  ele- 
ments, and  hence  some  contractors  have  an  idea  that  this  being 
the  case  almost  anything  can  be  utilized  as  aggregate.  Thus 
cinders  in  which  there  is  quite  a  large  percentage  of  ash,  partly 
burned  coal  and  the  like,  have  been  used  in  some  cases  and  with 
exceedingly  bad  results. 

In  one  case  where  the  concrete  had  been  made  from  cinders  from 
Southern  Iowa  coal,  the  concrete  after  it  was  cast  in  the  form  of 
slabs  expanded  to  such  an  extent  that  it  pushed  the  face  brick  out 
of  the  side  of  the  building  and  the  slabs  checked  and  cracked  to  a 
considerable  extent  owing  to  this  same  action. 

The  following  is  from  an  articles  by  Mr.  D.  B.  Butler  in  the 
Engineering  Record.* 

"EXPANSION    OF    CONCRETE    MADE    WITH    COKE 
BREEZE." 

"On  account  of  a  number  of  failures  of  roof  and  floor  slabs,  made 
of  coke  breeze  concrete,  which  were  called  to  his  attention,  Mr.  D.  B. 
Butler  undertook  a  series  of  experiments  to  determine  the  expansion 
of  such  concrete  since  an  examination  of  the  faulty  structures 
indicated  that  such  action  was  responsible  for  the  failures.  His 
conclusions  were  presented  in  a  paper  before  a  recent  meting  of 
the  Society  of  Architects, England,  from  which  these  notes  are  taken. 

"In  nearly  all  samples  of  so-called  breeze  concrete  examined  by 
Mr.  Butler,  a  very  considerable  quantity  of  material  other  than 
pure  coke  was  noticeable  in  the  aggregate,  such  as  clinkers,  stones, 
shale  and  ashes,  together  with,  in  some  instances,  a  noticeable 
amount  of  coal.  Whatever  may  be  the  disadvantages  of  other  ex- 
traneous material  found  in  breeze,  coal  is  not,  in  Mr.  Butler's  opinion 
a  desirable  constituent  for  concrete;  in  the  first  place,  on  account 
of  its  smooth,  shiny  surface,  the  adherence  of  the  cement  would  be 
extremely  poor;  in  the  second  place,  it  is  worse  than  useless  as  a 
fireproof  material,  an  account  of  its  tendency  to  decompose  on 
heating.  The  question  arose,  however,  whether  apart  from  being 
undesirable  for  the  reasons  aforesaid,  either  coke  breeze  or  coal  was 
in  any  way  dangerous  as  being  likely  to  cause  expansion  of  the 
concrete. 

"  The  first  experiment  was  of  a  somewhat  rough  and  ready  nature, 
and  was  made  with  coal.  An  ordinary  bituminous  house  coal  was 
crushed  and  sifted  about  the  fineness  of  standard  sand ;  with  this  coal 
a  1  to  3  mortar  was  made,  and  two  small  2-ounce  glass  bottles 
filled  with  the  mixture;  one  bottle  was  filled  quite  full,  and  the  other 
was  filled  to  within  a  quarter  inch  of  the  top  and  sealed  down  with 
a  paste  of  neat  cement,  the  object  of  the  sealing  being  to  ascertain 
whether  the  imprisonment  of  any  hydrocarbons  set  free  from  the  coal 

"June  19,  1909,  page  767. 


378  CINDER    CONCRETE 

would  have  any  bursting  effect.  For  comparative  purposes  similar 
bottles  were  also  filled  with  a  paste  of  neat  cement  and  1  to  3  mortar 
of  standard  sand. 

"The  whole  of  the  bottles  eventually  cracked,  with  the  exception 
of  one  filled  with  standard  sand-cement  mortar.  But  while  those 
entirely  filled  with  the  coal  mortar  generally  cracked  within  two  or 
three  days,  and  with  a  very  few  exceptions  continued  to  expand 
until  the  bottles  burst  away  into  several  pieces,  those  filled  with  the 
neat  cement  and  the  sand  mortar  frequently  did  not  develop  any 
cracks  whatever  till  several  months,  and  it  was  the  exception  rather 
than  the  rule  for  them  to  continue  expanding  sufficiently  to  burst 
the  bottle.  Both  the  neat  cement  and  sand  cement  mortar  bottles 
remained  perfectly  sound  after  eleven  months  and  then  only  develop- 
ed very  minute  cracks,  whereas  the  coal  mortar  bottle  was  cracked 
in  twelve  days  and  burst  right  off  in  forty  two  days.  This  suggests 
that  the  cause  of  the  cracking  after  such  protracted  periods  might 
be  due  to  unequal  expansion  of  the  glass  and  the  mortars  at  varying 
temperatures. 

"The  subsequent  experiments  were  made  with  rectangular 
bars  100  mm  long  and  22  mm  square  in  cross-section,  the  expansion 
and  contraction  of  which  were  accurately  measured  in  the  Bausch- 
inger  micrometer  calliper  apparatus.  By  the  use  of  this  a  minute  vari- 
ation of  0.005mm,  or  0.005  per  cent  in  the  length  of  the  prism,  may 
be  detected. 

"Eight  bars  or  prisms  were  made  with  satisfactory  cement,  four 
being  made  with  neat  cement  and  four  with  1  to  3  standard  sand 
cement  mortar.  Two  of  each  series  wrere  kept  entirely  in  air  and  two 
placed  in  water  after  twenty-four  hours  and  kept  therein  during 
three  months.  The  test  pieces  numbered  300  and  involved  5,000 
measurements. 

A  noticeable  feature  of  the  experiments  in  that  many  of  the 
specimens  which  show  very  marked  expansion  when  placed  under 
water  as  soon  as  set  expand  very  much  less  when  left  entirely  in  air. 
It  therefore  seemed  a  point  worth  determining  as  to  whether  ex- 
posure to  damp  or  moisture  would  in  any  way  affect  these  air-set 
specimens  at  the  end  of  the  three  months'  test,  after  they  had  be- 
come thoroughly  seasoned.  One  of  the  duplicate  air  bars  from 
each  series  was  therefore  placed  under  water,  the  time  elapsing 
between  the  date  of  moulding  and  placing  under  water  ranging  from 
91  to  292  days.  Immersion  had  practically  no  effect  upon  those 
specimens  which  had  previously  shown  no  expansion  which  kept 
under  water,  but  it  caused  almost  immediate  expansion  of  a  very 
serious  nature  with  those  fractions  of  breeze  which  had  previously  de- 
veloped expansion  when  placed  under  water  in  the  first  instance. 
This  clearly  shows  that  the  expansive  agent,  whatever  it  may  be, 
is  more  or  less  dormant  in  the  dry  air-set  block,  and  only  requires 
to  become  damped  to  constitute  a  serious  element  of  danger. 

"Taken  as  a  wrhole,  the  experiments  as  far  as  they  go  seem  to 
point  to  the  fact  that  as  regards  subsequent  expansion  there  is  not 
much  danger  to  be  apprehended  from  good,  cleam  coke  or  clinkers,  or 
or  even  anthracite  coal,  but  that  some  kinds  of  ashes  and  furnace 
refuse  are  highly  dangerous,  while  any  considerable  quantity  of 
bituminous  coal  is  absolutely  fatal.  One  noticeable  feature  of  the 
experiments,  however,  was,  that  most  of  the  coke-breeze  mortars 
had  a  tendency  more  or  less  seriously  to  attack  the  iron  moulds, 
causing  them  to  rust  during  the  short  space  of  twenty-four  hours  be- 
tween the  moulding  of  the  specimens  and  their  removal  from  the 
moulds.  Mr.  Butler  is  unaware  if  such  results  have  been  found 
to  any  appreciable  extent  in  actual  practice,  but  samples  of  breeze 
concrete  sent  him  for  examination  a  short  time  ago  showed  distinct 
marks  of  considerable  rusting  having  taken  place  where  the  concrete 
had  been  in  contact  with  the  rolled  joists." 


TAMPED    CONCRETE  379 

Mr.  Butler's  experiments  quoted  above  coincide  with  our  obser- 
vations. In  general,  cinder,  if  fit  to  use,  should  be  free  from  ash 
and  should  be  well  burned  stoker  clinker.  Concrete  made  from 
good  hard  clinker  has  proved  a  good  and  substantial  fire  proofing 
material. 

There  is  this  difficulty  in  its  use,  however,  that  the  contractor 
too  frequently  furnishes  cinders  rather  than  hard  clinker. 

3.  Concrete   Mixed   Dry   and   Tamped.     Concrete  mixed  dry 
and  tamped  in  the  old  fashioned  way  is  more  or  less  porous,  and  liable 
to  disintegrate  under  severe  conditions  of  exposure,  as  for  example, 
whenever    the    concrete  is  soaked  with  water,  frozen  and  thawed 
repeatedly.     Such  conditions  may  occur  in  an  aggravated  form  in 
retaining  walls. 

The  government  sea  wall  at  the  ship  canal,  Duluth,  made  in 
the  old  fashioned  manner,  mixing  the  concrete  dry  and  tamped  is 
showing  the  effect  of  exposure  to  a  far  greater  degree  than  we  should 
expect  had  the  work  been  executed  in  accordance  with  the  present 
standard  practice. 

In  this  sea  wall  it  should  be  noted  that  in  the  cold  season  the 
wall  was  alternately  wet  and  dry  as  the  waves  washed  against  it; 
that  moisture  is  alternately  frozen  and  thawed  in  the  exposed 
surface,  and  owing  to  the  fact  that  the  method  of  mixing  leaves  the 
concrete  slightly  porous  some  disintegration  naturally  results. 

In  general,  the  best  concrete  to  withstand  such  severe  conditions 
is  that  which  is  most  dense,  is  strongest,  and  made  from  the  hardest 
and  most  durable  stone  as  an  aggregate  and  with  clean,  coarse  sand. 

Where  brick  or  building  stone  is  made  of  a  fairly  dry  or  moist 
mixture  and  is  not  exposed  to  the  severe  conditions  above  described 
it  proves  very  durable  material. 

4.  Hair   Cracks,    Map   Checks  and   Crazing.     In  troweling   a 
finished  surface  on  concrete  the  moisture  is  brought  to  the  surface 
by  the  working  of  the  material,   resulting  in   somewhat   unequal 
conditions  of  moisture.     The  exposure  promotes  the  rapid  drying 
out  of  the  surface  and  causes  what  is  known  as  hair  cracks,  map 
checks  and  the  like.     These  are  generally  only  of  very  slight  depth 
and  mean  little  as  to  the  permanence  of  the  material,  providing  the 
concrete  is  made  of  good  cement  and  a  first  class  aggregate  is  used. 

A  peculiar  fact  concerning  this  defect  in  concrete  finished  sur- 
faces is  that  on  some  blocks  it  will  not  appear  at  all,  while  others 


380  CRAZING 

made  under  almost  identical  conditions  will  be  badly  affected. 
Perhaps  the  difference  in  part  may  be  accounted  for  by  the  thoro- 
ness  with  which  the  concrete  has  been  mixed,  the  time  expended  in 
mixing,  as  well  as  the  conditions  of  drying  and  curing. 

Concrete  which  has  been  thoroly  mixed  in  a  machine  for  double 
or  triple  the  ordinary  time  will  be  a  little  stronger  than  concrete 
which  has  been  mixed  for  only  fifteen  or  twenty  revolutions.  If 
the  mixing  is  continued  for  twenty  minutes  there  will  be  less  tend- 
ency towards  rapid  setting  and  shrinkage  and  the  development  of 
checks  and  cracks,  much  on  the  order  of  the  results  obtained  by 
skilled  mechanics  by  retempering  cement  mortar  in  patching  old 
work. 

In  the  treatment  of  concrete  which  is  finished  with  a  troweled 
.surface  to  prevent  checking  it  is  desirable,  where  it  is  exposed,  to 
keep  it  protected  by  burlap  soaked  in  water  and  to  keep  the  direct 
rays  of  the  sun  from  it  by  an  additional  cover  of  canvas.  In  this 
way  steps  and  similar  work  may  be  executed  with  the  least  trouble 
from  this  source,  provided  care  has  been  used  in  the  selection  of  both 
sand  and  stone  used  as  aggregates. 

In  the  manufacture  of  cast  stone  this  difficulty  is  one  which  the 
worker  in  this  field  is  forced  to  meet  and  it  is  essential  that  the 
mixing  should  continue  without  intermission  until  the  material  is 
run  into  the  mold. 

In  general,  cast  stone  made  by  the  sand  mold  process  will  keep 
its  color  better  than  such  natural  stone  as  Bedford,  altho  it 
may  discolor  in  streaks  and  blotches  known  as  crazing.  Efforts 
made  to  overcome  crazing  may  be  summarized  as  follows: 

First,  by  the  addition  of  other  ingredients  to  the  cement  in 
mixing  with  the  intent  to  render  the  material  more  perfectly  water- 
proof and  more  uniform  in  setting. 

Second,  to  coat  or  waterproof  the  material  after  it  has  been 
cast,  with  a  compound  repellent  to  moisture. 

Third,  to  remove  a  thin  layer  of  the  surface  of  the  stone  and 
concrete  and  get  below  the  depth  of  the  hair  lines  or  depressions 
which  form  in  casting  and  cause  this  peculiar  marking  or  discolora- 
tion when  exposed  to  the  weather. 

The  first  two  methods  have  apparently  been  successful  in  some- 
what mitigating  this  difficulty,  while  the  third  method  has  been 
successful  as  practiced  by  the  Roman  Stone  Company,  of  Toronto. 
Their  method  is  to  use  a  carborundum  wheel,  dressing  and  tooling 
the  surface  therewith. 


EFFECT    OF    OIL,    GREASE,    ETC.  381 

5.  Temperature  Effects.     Changes  of  temperature  cause  changes 
in  the  volume  in  concrete    as  in  all  materials  with  which  we  have 
to  deal.     The  difficulty  which  is  encountered  is  the  cracking  of  the 
concrete  as  it  is  brought  into  tension  by  change  in  volume.     Massive 
walls,  unless  cut  at  intervals  of  thirty  feet  or  thereabouts  will  crack 
thru  from  this  cause.      Where  openings,  such  as  windows,  are  cut 
thru  a  solid  wall  of  concrete  cracks  are  liable  to  develop  at  the  corners 
unless  the   concrete  is  well  reinforced  by  steel   rods  crossing  the 
corners  in  such  a  manner  as  to  take  care  of  this  tension  and  prevent 
the  development  of  cracks. 

In  slabs  reinforced  in  one  direction  there  should  be  used  not 
less  than  eight-hundredths  percent  of  metal  for  temperature  stress 
if  it  is  expected  to  prevent  the  development  of  unsightly  checks. 

6.  Disintegration  of  Concrete  by  Oil,  Grease,  etc.     In  factory 
buildings,  machine  shops,  etc.,  oil  and  grease  are  liable  to  come  in 
contact  with  the  concrete  and  it  is  important  to  know  what  effect 
it  will  have  upon  the  material.     Certain  kinds  of  oils  are  known  to 
be  positively  injurious  to  concrete  in  the  earlier  stages  of  hardening 
and  to  disintegrate  it  to  a  considerable  extent.     Where,  however, 
the  concrete  has  had  ample  time  to  harden  there  seems  to  be  little 
if  any  damage  resulting  from  lubricating  oils  such  as  are  ordinarily 
employed  in  a  factory  or  machine  shop.     Where  it  is  desired  to  use 
a  floor  which  has  not  had  at  least  two  months  in  which  to  thoroly 
harden  we  would  recommend  coating  the  concrete  with  some  good 
waterproofing  compound  or  floor  paint,  thereby  protecting  it  until 
after  it  has  had  opportunity  to  become  thoroly  cured  and  hardened 
thruout . 

The  question  of  disintegration  of  Portland  cement  briquettes 
and  experiments  to  prevent  it  have  been  quite  fully  discussed  by 
Mr.  James  D.  Hain,  Assoc.  M.  Am.  Soc.  C.  E.,  in  the  Engineering 
News,  March  16,  1005.  His  conclusions  may  be  summarized  as 
follows : 

1.  Most    oils   penetrate    concrete    mortar,    which   makes   them 
dangerous. 

2.  Concrete  is  more  liable  to  be  disintegrated  when  saturated 
with  oils  and  fats  if  not  thoroly  set. 

3.  A  good  quality  of  concrete  is  less  liable  to  be  damaged  by 
oil  than  a  poorer  quality,  such  as  a  porous,  poorly  mixed  or  im- 
properly seasoned  concrete. 

4.  Ordinary   concrete   work    is   rarely    subjected   to    continued 


382 


HAIN'S    EXPERIMENTS 


doses  of  oil.  It  is  more  often  only  occasionally  spattered.  Dis- 
integration under  the  latter  conditions  seems  remote,  especially 
in  the  case  of  the  first  class,  well  seasoned  concrete. 

Last,  even  tho  subjected  to  the  equivalent  of  continued  satura- 
tion, this  disintegration  would  be  long  drawn  out  if  the  concrete 
were  properly  made  and  well  set.  Even  under  ordinary  conditions 
it  seems  desirable  to  use  a  wash  for  oil  spattered  concrete  to  pre ven  t 
the  oil  from  penetrating  it. 

Mr.  Hain  in  his  experiments  tried  the  following  wash:  Five 
per  cent  solution  of  alum  and  a  seven  percent  solution  of  castile 
soap,  and  also  experimented  with  paraffine.  None  of  these  proved 
satisfactory  where  the  briquettes  were  immersed  in  oil. 

The  following  table  shows  the  result  of  some  of  these  experi- 
ments of  Mr.  Hain: 


Extract 

Whale 

Castor      Linseed      Petro- 

Signal 

No.  Bri- 

Class of              Mixture 

Lard  Oil 

Oil 

Oil              Oil        leum  Oil 

Oil 

quettes 

Portland            Portland 

(Crude) 

made 

Cement              Cement 

,                                     ' 

and  Sand 

Time  applied  before  disintegration 

18 

Stone  and  clav 

Neat  

3  mos.  . 

*                 *                                                      * 

12 

Stone  and  clav 

1:3  sand.  .  . 

* 

*                 *                                                      * 

18 

Marl  and  clav. 

Neat  

21  mos. 

*                 *                                               6     mos 

12 

Marl  and  clay. 

1:3  sand.  .  . 

* 

*       ;          *                                                      * 

18 

Slag  and  stone 

Neat  

1  mo.  .  . 

3  mos.  . 

4  mos.  .  i                                        li  mos 

12 

Slag  and  stone 

1:3  sand..  . 

7  mos.  . 

4i  mos. 

6J  mos. 

4    mos 

"Sound  after  applying  oil  nine  months  at  which  tests  were  discontinued . 
All  briquettes  set  seven  days  in  air  before  applying  oil. 


Mr.  Reid  in  his  work  on  concrete  states  that  one  of  the  briquettes 
tested  with  signal  oil  was  sent  to  the  laboratory  of  Toch  Brothers, 
Long  Island  City,  and  a  careful  analysis  was  made.  Mr.  Maximilian 
Toch  states  that  a  determination  of  the  soluble  substances  in  the 
briquette  showed  that  the  disintegration  was  due  to  the  formation 
of  oleate  and  stearate  of  calcium.  To  reduce  this  to  its  simplest 
expression,  the  animal  oils  contain  acids  which  combine  with  the 
lime  and  crystals  and  stearate  and  oleate  of  lime  are  formed.  It 
is  very  likely  that  these  crystals  in  the  process  of  formation  have 
increased  in  bulk  in  the  briquette  and  the  bond  which  has  been 
formed  by  the  lime  in  the  set  cement  has  been  totally  disintegrated 
and  ruptured.  These  crystals  were  isolated  and  verified  under  the 
microscope. 

Mr.  Toch  also  states  that  machine  oils  are  almost  all  paraffine 
oils,  do  not  contain  animal  fats,  and  hence  do  not  affect  concrete. 


INJURY    BY    ELECTROLYSIS  383 

Silicate  of  magnesia,  sold  under  the  name  of  fluate,  has  often 
been  used  as  a  wash  to  protect  concrete  against  the  action  of  oil. 
When  this  wash  is  applied  to  concrete,  silica  is  liberated  and  fills 
up  the  pores.  The  magnesium  fluate  acts  as  a  binder,  and  the  ce- 
ment becomes  excessively  hard  after  a  few  months.  Limestone 
and  building  stone  have  been  treated  with  this  material  in  Europe 
with  great  success.  This  compound  is,  however,  expensive. 

7.  Disintegration  of  Reinforced  Concrete  by  Electrolysis.  Labor- 
atory experiments  by  Toch,  Knudson  and  Langsdorf  in  1906  and  1907 
show  that  under  certain  circumstances  passage  of  direct  electric 
current  from  the  reinforcing  metal  into  the  concrete  gives  rise  to 
corrosion  of  the  metal  and  to  cracking  and  splitting  of  the  surround- 
ing concrete  which  seems  to  be  brought  about  by  the  mechanical 
pressure  developed  by  the  oxides  which  occupy  a  volume  over  twice 
as  great  as  the  metal  from  which  they  are  formed. 

The  conditions  under  which  reinforced  concrete  may  be  seriously 
injured  by  electrolysis  are  fortunately  exceptional  rather  than  the 
rule.  That  it  may  be  injured  even  in  exceptional  cases  presents  a 
problem  of  importance  which  requires  a  statement  of  the  conditions 
under  which  injury  may  occur  and  the  method  of  controlling  them. 

The  conditions  under  which  electrolysis  may  occur  and  the 
concrete  suffer  by  electrolysis  are  moisture  and  difference  of  potential 
between  the  electrodes  and  contact  with  the  mass  of  the  concrete. 
Perfectly  dry  concrete  below  grade  level  is  seldom  found,  while  there 
are  few  places  in  our  cities  at  the  present  time  where  some  appreciable 
differences  of  potential  cannot  be  found  between  two  points  which 
are  more  than  a  few  yards  apart.  On  the  other  hand,  concrete  has 
to  be  very  wet  in  order  to  possess  a  maximum  of  conductivity.  Any 
reduction  of  the  moisture  content  below  the  saturation  point  causes 
an  increase  in  its  resistance  and  consequent  decrease  in  the  current 
which  will  flow  thru  the  concrete  under  a  given  potential  difference. 

The  resistance  of  ordinary  air-dried  concrete  is  usually  about  ten 
times  that  of  wet  concrete,  and  for  this  reason  concrete  above  grade 
level  is  less  susceptible  to  electrolitic  damage  than  if  located  where 
it  is  permanently  wet.  While  air-dried  concrete  is  not  immune 
from  electrolysis  troubles,  difference  of  potential  due  to  stray 
currents  is  rarely  sufficient  to  produce  trouble  and  in  the  absence 
of  special  conditions  electrolitic  damage  to  concrete  at  any  level 
above  grade  is  extremely  rare. 

Special  emphasis  should  be  laid  upon  the  conditions  that  are 
liable  to  produce  damage  by  the  flow  of  the  electric  current  between 


384  STRAY    CURRENTS 

electrodes  in  contact  with  the  concrete.  The  conduction  being 
electrolitic  the  reactions  take  place  only  at  the  electrodes  and  in 
the  absence  of  such  electrodes  no  reactions  occur  within  the  concrete. 
The  only  effect  therefore  would  be  the  slow  removal  of  the  consti- 
tuents which  are  soluble  in  water  and  hence  the  effect  on  plain  con- 
crete would  not  be  essentially  different  from  that  of  slow  water 
seepage. 

Sources  of  Stray  Currents.  The  sources  of  potential  differences 
in  concrete  structures  may  be  classified  under  two  heads:  those 
due  to  direct  contact  between  conductors  of  direct  current  lighting 
or  power  circuits  in  some  part  of  the  building  and  those  which  have 
their  origin  in  stray  currents  from  electric  railways  or  other  grounded 
power  lines. 

The  former  may  happen  in  any  building  containing  direct  current 
electric  wires  thru  defective  insulation.  It  is  not  necessary  that 
both  sides  of  the  line  be  grounded  in  the  building  itself,  since  if  one 
side  of  the  line  is  grounded  on  the  building  and  the  other  grounded 
in  some  remote  quarter  of  the  system  those  portions  of  the  building 
near  the  wire  may  be  subjected  to  a  considerable  difference  of 
potential.  If  the  wire  be  grounded  directly  on  the  concrete  and 
not  on  the  reinforcement  the  comparatively  small  section  of  the 
path  of  the  current  near  the  point  of  contact  between  the  concrete 
and  the  wire  will  cause  most  of  the  total  drop  of  potential  to  the  ground 
to  occur  within  the  restricted  region  near  the  wire  and  it  is  only 
here  that  any  damage  may  be  expected,  and  since  the  current  will 
be  small  the  damage  if  any  will  be  small. 

Ultimately  in  case  both  sides  of  the  line  are  not  grounded  in  the 
structure  any  current  that  leaks  from  the  wire  would  pass  into  the 
earth  thru  the  footings  and  foundations  and  thru  pipe  systems 
entering  the  building.  The  cross  section  of  these  paths  is  so  large 
in  the  aggregate  that  the  potential  gradients  would  be  insufficient 
to  raise  the  temperature  appreciably  and  no  appreciable  damage 
is  likely  to  occur,  since  the  corrosion  above  referred  to  occurs  in  wet 
concrete  only  and  not  to  any  considerable  extent  until  at  least  a  tem- 
perature of  113°  Fahr.  is  reached.  According  however,  to  determina- 
tions made  by  the  Bureau  of  Standards,  if  the  power  wire  be  grounded 
directly  on  a  portion  of  the  reinforcing  metal  the  condition  is  more 
serious.  The  extent  of  the  damage  will  be  greater  if  such  large 
area  of  the  reinforcement  is  in  metallic  contact  with  the  electric 
wires,  as  to  reduce  the  resistance  across  this  area  to  a  small  amount. 
In  case  the  ground  is  on  the  positive  side,  the  potential  gradient  near 
the  reinforcement  may  become  high  enough  to  cause  rapid  corrosion 


EXPERIMENTS    ON    CORROSION  385 

and  consequent  destruction  of  the  reinforcing  metal.  If  on  the 
other  hand  the  reinforcing  metal  be  the  negative  electrode  a  softened 
condition  of  the  concrete  would  be  developed  near  the  surface  of  the 
iron  which  would  tend  to  destroy  the  bond  and  this  would  probably 
be  the  more  serious  condition  of  the  two  since  it  will  not  manifest 
itself  by  producing  local  cracks  in  the  concrete  and  might  not  be- 
come known  until  a  large  portion  of  the  building  had  become 
weakened. 

While  such  a  condition  as  this  might  possibly  occur,  and  if 
neglected  might  become  serious,  it  is  nevertheless  a  trouble  that 
can  be  readily  guarded  against. 

The  other  source  of  current  that,  under  certain  circumstances, 
might  possibly  give  rise  to  trouble  is  the  grounded  current  of  electric 
railways.  Various  electrolysis  surveys  show  that  a  potential  dif- 
ference exceeding  two  volts  due  to  stray  currents  between  any  two 
parts  of  the  building  is  extremely  rare  and  this  would  almost  in- 
evitably be  distributed  over  so  great  a  distance  that  the  potential 
gradient  would  not  be  great  enough  to  cause  appreciable  trouble. 
Stray  currents  may  enter  a  building  thru  water  pipes,  gas  pipes,  lead 
cable  sheathes  and  the  like.  Differences  of  potential  considerably 
larger  than  two  volts  may  be  brought  about  between  the  different 
portions  of  the  building  in  this  manner  or  between  parts  of  the 
building  and  the  earth.  If  the  pipe  systems  come  in  contact  with 
the  concrete  only  and  not  with  the  reinforcing  metal  any  damage  that 
may  occur  will  be  slight  and  confined  to  the  immediate  vicinity  of  the 
pipes  or  cables,  but  if  the  pipes  come  into  metallic  contact  with  the 
reinforcement  the  latter  comes  to  the  same  potential  as  the  pipes  and 
may  become  anode  or  cathode  according  to  the  direction  of  flow  in 
the  circuit.  Cases  may  arise  where  a  difference  of  potential  of 
serious  magnitude  may  be  produced  in  this  way. 

Conclusions  from  Laboratory  Experiments.  Laboratory  experi- 
ments show  that  the  corrosion  of  iron  even  in  wet  concrete  is  very 
slight  at  temperatures  below  113  degrees  Fahr.  For  any  fixed 
temperature  the  amount  of  corrosion  for  a  given  number  of  ampere 
hours  is  independent  of  the  current  strength.  The  rapid  destruction 
of  anode  specimens  of  moist  concrete  at  high  voltage  (60  to  100 
volts  or  more)  is  made  possible  mainly  by  the  heating  effect  of  the 
current  which  raises  the  temperature  above  the  limit  above  stated. 
If  the  specimens  be  artifically  cooled  no  appreciable  corrosion  occurs 
and  no  cracking  results.  The  potential  gradient  necessary  to  produce 
a  temperature  rise  to  113  degrees  Fahr.,  with  consequent  corrosion 
was  60  volts  per  foot  in  the  specimens  tested  by  the  U.  S.  Bureau 


386  PRECAUTIONS 

of  Standards.  For  air  dried  concrete  it  is  much  higher.  This  in- 
dicates that  under  actual  conditions  corrosion  caused  by  stray 
currents  may  be  expected  only  under  very  unusual  and  special 
conditions. 

Specimens  of  normal  wet  concrete  carrying  currents  increase 
their  resistance  100  fold  or  more  in  the  course  of  a  few  weeks  which 
fact  further  lessens  danger  of  trouble.  The  presence  of  a  small 
amount  of  salt  greatly  increases  the  initial  conductivity  of  wet  con- 
crete thus  allowing  more  current  to  flow  and  it  also  destroys  the 
passive  conditions  of  iron  at  ordinary  temperatures  increasing  the 
rate  of  corrosion  and  consequent  tendency  of  the  concrete  to  crack. 

Concrete  structures  built  in  contact  with  salt  water  or  in  salt 
marshes  are  more  susceptible  to  electrolysis  than  concrete  not  sub- 
jected to  such  influences. 

Conditions  may  arise  in  practice  which  give  rise  to  damage  due 
to  stray  currents  but  the  danger  from  this  source  has  been  greatly 
over-estimated.  While  precautions  are  necessary  under  certain 
conditions  there  is  no  cause  for  serious  alarm. 

It  may  be  here  noted  that  alternating  currents  have  largely 
displaced  direct  currents  for  lighting  and  power  purposes  because  of 
reduced  transmission  losses  in  alternating  circuits.  It  should  be 
further  noted  that  office  buildings  with  structural  steel  skeletons 
which  have  passed  thru  the  period  during  which  direct  current  was 
used  almost  entirely  for  lighting,  have  suffered  little  or  not  at  all 
from  this  cause  and  it  may  be  stated  that  there  is  little  or  no  reason 
to  expect  electrolysis  trouble  in  reinforced  concrete  buildings  gener- 
ally, and  that  only  in  special  classes  of  buildings  such  as  ice  cream 
factories,  cold  storage  plants,  packing  houses,  and  the  like,  where 
steam  and  ammonia  or  acid  fumes  come  in  contact  with  the  floors 
may  serious  trouble  be  expected  from  this  cause. 

Protective  Measures.  In  all  cases  it  is  conservative  to  forego 
the  use  of  salt  or  chloride  of  calcium  in  winter  on  reinforced  con- 
struction below  grade  level  regardless  of  the  character  of  the  building 
This  would  demand  greater  care  in  protecting  the  work  and  heating 
material  for  this  part  of  the  structure.  Also  careful  attention  should 
be  given  to  the  insulation  of  gas  pipes,  water  pipes  and  soil  pipes 
from  all  contact  with  the  reinforcement.  Proper  insulation  of  the 
wires  should  be  provided  so  that  any  leakage  to  reinforcement  may 
be  prevented.  Finally  in  that  class  of  buildings  in  which  the  con- 
ditions are  favorable  to  damage  from  electrolysis,  alternating  cur- 
rents for  lighting  and  power  should  preferably  be  adopted.  In 
bridge  work  for  carrying  electric  lines  and  power  lines  fiber  conduits 


RECOMMENDATIONS  387 

are  to  be  preferred  to  metal  conduits.  Any  possible  damage  from 
alternating  currents  may  be  considered  insignificant  or  negligible 
in  comparison  to  that  by  direct  currents  transmitting  the  same  power. 
Mr.  H.  P.  Brown,  Engineering  News,  June,  1911,  offers  the 
following  suggestions  in  regard  to  electrical  currents  in  damp  rein- 
forced concrete  buildings: 

Do  not  depend  for  insulation  upon  even  the  best  rubber  covered 
wire  nor  upon  japanned  conduits  in  rooms  subjected  to  fumes 
and  vapors. 

Do  not  permit  the  grounding  of  the  intermediate  wire  in  three- 
wire  systems. 

Do  not  permit  any  grounding  of  the  secondary  circuit  of  a 
transformer. 

In  the  vicinity  of  electric  power  houses  or  substations  use 
wooden  pipe  for  gas  or  water  serving  pipes  from  the  street  mains. 
Use  insulating  tubes  around  the  gas,  water  or  steam  pipes  where 
they  pass  thru  concrete  floors  or  walls. 

The  following  are  recommendations  from  Bulletin  No.  18,  U.  S. 
Bureau  of  Standards  on  "Electrolysis  in  Concrete"  by  E.  B.  Rosa, 

Burton  McCollum,  and  O.  S.  Peters. 

In  order  to  insure  safety  of  reinforced  concrete  from  electrolysis 
the  investigation  shows  that  potential  gradients  must  be  kept 
much  lower  in  structures  exposed  to  the  action  of  salt  waters, 
pickling  baths,  and  all  solutions  which  tend  to  destroy  the  passive 
state  of  iron. 

All  direct  current  electric  power  circuits  within  the  concrete 
building  should  be  kept  free  from  grounds.  If  the  power  supply 
comes  from  a  central  station  the  local  circuits  should  be  periodically 
disconnected  and  tested  for  grounds  and  incipient  defects  in  the 
insulation.  In  the  case  of  isolated  plants  ground  detectors  should 
be  installed  and  the  system  kept  free  from  grounds  at  all  times. 

All  pipe  lines  entering  concrete  building  should,  if  possible, 
be  provided  with  insulating  joints  outside  the  building.  If  a  pipe 
line  passes  thru  a  building  and  continues  beyond,  one  or  more 
insulating  joints  should  be  placed  on  each  side  of  the  building.  If  the 
potential  drop  around  the  isolated  section  is  large,  say,  8  or  10 
volts  or  more,  the  isolated  portion  should  be  shunted  by  means  of 
a  copper  cable. 

Lead-covered  cables  entering  such  buildings  should  be  isolated 
from  the  concrete.  Wooden  or  other  nonmetallic  supports  which 
prevent  actual  contact  between  the  cable  and  the  concrete  will 
give  sufficient  isolation  for  the  purpose.  Such  isolation  of  the  lead- 
covered  cable  is  desirable  for  the  protection  of  the  cable  as  well  as  the 
building. 

Partial  Bibliography  of  Electrolysis  in  Concrete: 

Max  Toch,  German  Electric  Chemical  Society  Vol.  9,  page  77,1906 

"       Iron  Trade  Review,  Vol.  46,  page  1007,  1910. 
Engineering  News,    Vol.  60,  page  710,  A.  J.  Nicholes. 
Vol.  63,  page  372,  O.  L.  Eltinge. 
Vol.  65,  page  684,  H.  P.  Brown. 
Vol.  66,  page  10,  Barker  &  Upson. 
Vol.  66,  page  152,  B.  C.  Worth— H.  P.  Brown. 
Vol.  66,  page  207. 

Engineering  Record,  Vol.  62,  page  132. 
Vol.  63,  page  272. 

U.  S.  Bureau  of  Standards,  Bull.  No.  18,  Electrolysis  in  Concrete. 
Am.  Electric  Ry.  Practice,  Herrich  &  Boynton,  (Electrolysis  and 
its  Prevention),  page  387,  397. 


388 


CHAPTER  XIII 

1.  Floor  Finish.     The  rough  concrete    slab   in   a   warehouse, 
factory  or  office  building  may  be  finished  in  a  variety  of  ways.     The 
requirements  for  factory  purposes  and  office  buildings  frequently 
demands  a  wood  finished  floor,  while  in  a  warehouse  and  some  classes 
of  manufacturing  buildings,  the  concrete  finished  floor  is  preferable. 

2.  Strips  and  Strip  Fill  for  Wood  Floors.     Preparation  of  the 
rough  slab  is  made  for  the  wood  floor  as  follows:     Parallel  strips 
are  laid  at  about  sixteen  inch  intervals,  embedded  in  concrete  and 
the  flooring  nailed  thereto.     The  better  time  to   apply  the  strip 
fill  is  immediately  after  the  rough  slab  is  sufficiently  hardened  to 
work  upon  for  the  reason  that  at  this  time  the  strips  can  be  readily 
spiked   down   to   the   partly   hardened   concrete   and   wedged   and 
aligned  to  the  proper  level  without  difficulty.     Then  the  strip  fill 
can  be  put  in  with  the  same  equipment  that  has  been  used  to  cast 
the  floor  slab.     For  the  reason  that  a  hard  strip  fill  adds  materially 
to  the  strength  and  stiffness  of  the  slab  it  is  preferable  to  make  the 
mix  approximately  the  same  as  that  for  the  slab  except  where  the 
loads  are  light  and  strength  is  no  object.     Then  a  1  :  3j  :  4  mix  is 
ample  for  all  purposes.     In  this  mix,  however,  the  coarser  aggregate 
should  be  in  the  form  of  gravel  in  size  from  f  inch  down,  or  crushed 
stone  from  J  inch  down. 

No  natural  cement  or  lime  should  be  used  in  the  mix  since  where 
it  is  used  trouble  almost  inevitably  results  by  reason  of  its  slow  hard- 
ening. The  moisture  in  the  strip  fill  swells  and  expands  the  wood 
flooring  to  such  an  extent  that  it  springs  away  from  the  fastenings 
to  the  strips  necessitating  the  entire  relaying  of  the  floors  in  many 
cases. 

Strips  are  economically  and  conveniently  made  by  splitting  up 
old  centering  lumber,  such  as  4  by  4s,  ripping  them  thru  the  center 
and  then  ripping  the  2  by  4s  with  a  beveled  cut  giving  a  strip  1J 
inches  wide  at  top,  2J  inches  wide  on  the  bottom  and  If  inches  deep. 
This  is  a  good  way  to  work  off  the  old  lumber. 

3.  Width  of  Flooring.     Narrow  widths  of  maple  flooring  are 
preferable  to  the  wider  widths.     Two  and  one-half  inches  is  as  wide 


CEMENT   FINISHED    FLOORS  389 

as  can  be  recommended  where  the  floor  is  J  inches  in  thickness. 
Where  If  or  If  thickness  is  used  3J  to  3J  inches  is  the  preferable 
limit. 

4.  Cement  Finish  Coat.  In  no  part  of  concrete  construction 
has  there  been  so  much  difficulty  in  securing  first  class  and  satis- 
factory work  as  in  putting  down  concrete  floor  finish.  A  good  bond 
is  desired  between  the  finish  coat  and  the  concrete  of  the  rough  slab. 
To  secure  this  bond,  some  endeavor  to  apply  the  finish  integrally 
with  the  rough  slab.  The  difficulty  attending  this  method  is  the 
shrinkage  of  the  body  of  the  work,  checking  and  injuring  the  top 
coat  together  with  the  fact  that  if  this  finish  is  applied  while  the  rough 
slab  is  still  plastic  the  hardening  of  the  surface  in  getting  its  initial 
set  will  be  slow.  This  means  that  some  partly  hardened  cement 
will  be  broken  up  in  troweling  and  the  finish  will  be  brittle  and  will 
dust  badly  altho  it  looks  well  when  the  work  is  first  completed. 

Again,  working  on  the  finish  before  it  has  sufficient  time  to  dry 
in  placing  the  centering  thereon  for  the  succeeding  stories  is  quite 
likely  to  scratch  and  mar  the  finish  and  leave  it  in  bad  shape  when 
the  building  is  done. 

A  further  difficulty  occurs  from  unequal  moisture  conditions 
about  the  base  of  the  column.  When  the  columns  for  the  next  story 
are  cast  the  excess  of  water  in  the  mix  wets  down  and  swells  the 
concrete  surface  about  their  bases  and  expands  it  and  as  this  dries 
more  slowly  than  the  rest  of  the  slab  shrinkage  checks  and  spider 
web  cracks  will  very  likely  be  found  in  the  finish  about  the  column 
bases  when  it  dries,  if  the  work  has  been  executed  in  this  manner. 

The  application  of  the  finish  coat  before  casting  the  succeeding 
story  has  this  advantage:  The  dripping  from  the  floor  above 
does  not  coat  the  rough  slab  and  prevent  securing  a  good  bond 
thereto,  as  sometimes  happens  where  this  dirt  and  cement  wash  is 
not  removed  before  the  application  of  the  finish.  The  trouble 
of  cleaning  the  floor  thoroly  is  thereby  obviated. 

If  it  is  desired  to  lay  the  finish  before  carrying  up  the  next  story, 
it  is  recommended  that  after  the  concrete  has  stiffened  up  and  before 
it  is  thoroly  hard,  the  surface  be  roughened  with  a  rake  to  secure 
a  better  bond  for  the  finish,  and  that  the  rough  slab  be  then  allowed 
to  stand  for  not  less  than  twenty-four  hours  in  good  drying  weather 
and  longer  where  the  weather  is  chilly  so  that  the  rough  slab  be- 
comes thoroly  rigid.  Then  the  finish  may  be  successfully  applied, 
cutting  it  back  from  the  foot  of  the  column  in  squares  at  least  a 
foot  from  the  column  base,  and  applying  this  part  of  the  finish  at 


390  HARDENING    COMPOUNDS 

a  later  date.  The  surface  then  should  be  protected  preferably  by 
sawdust  thoroly  wet  down,  while  the  centering  for  the  floors  above 
should  be  well  supported  on  planks  so  that  the  finish  coat  will  not 
be  abraded  during  the  process  of  curing. 

If  the  finish  is  to  be  applied  after  the  rough  work  of  the  building 
is  complete,  which  is  the  usual  manner,  the  surface  of  the  slab  should 
be  first  thoroly  cleaned  of  dirt,  and  the  drippings  from  the  upper 
floor  removed.  It  should  then  be  prepared  to  receive  the  finish, 
by  thoroly  soaking  with  water. 

5.  The  Mix  of  Finish.     The  mix  of  the  finish  should  preferably 
be  one  part  Portland  cement,  to  one  and  one-half  parts  clean  coarse 
sand.     A  good  siliceous  sand  is  a  better  aggregate  for  this  purpose 
than  crushed  granite. 

The  finish  coat  should  be  thoroly  mixed  as  a  stiff  paste,  screeded 
to  the  proper  level  as  it  is  applied,  and  as  soon  as  it  has  taken  its 
initial  set,  troweled  and  rubbed  to  a  smooth  surface. 

The  cause  of  much  of  the  difficulty  with  floor  finish  is  due  to  the 
mistaken  idea  that  it  requires  a  very  wet  mix  to  secure  a  good  bond 
to  the  old  concrete.  With  this  sloppy  mix  more  or  less  separation 
occurs  and  the  inert  material  and  laitance  comes  to  the  surface. 
Then  when  it  is  leveled  off  the  workmen  are  obliged  to  wait  until 
some  of  the  cement  has  attained  its  final  instead  of  its  initial  set 
before  they  can  proceed  to  trowel  off  the  finished  surface.  The  ce- 
ment in  the  finish  that  has  attained  its  final  set  is  broken  up  and  does 
not  recover  its  strength  while  that  which  has  not  progressed  so  far 
hardens  in  a  normal  manner.  The  result  is  that  the  portion  of  the 
cement  which  has  attained  its  final  set  combined  with  inert  material 
brought  to  the  surface  by  troweling  forms  a  dust  which  is  readily 
rubbed  up  on  the  finish.  The  condition  of  the  floor  finished  in  this 
wise  is  better  or  worse  dependent  on  the  following  conditions : 

Where  temperature  conditions  are  such  that  the  cement  hardens 
very  slowly,  as  in  the  fall  of  the  year,  and  the  finish  is  allowed  to  stand 
five  or  six  hours  before  it  gets  sufficiently  hard  to  work  upon,  the 
resulting  finish  is  most  inferior.  Where,  however,  the  temperature 
conditions  are  such  that  the  cement  sets  more  rapidly,  a  much  better 
surface  results,  sometimes  one  that  is  fairly  satisfactory. 

6.  Hardening  Compounds.     A  number  of  compounds  have  been 
placed  on  the  market  to  harden  floor  finish  and  render  it  tougher 
under  wear.     Good  results  with  any  of  these  compounds  depend, 
as  in  the  case  of  the  cement  finish,  upon  proper  workmanship  and 


CONCRETE    STAIRS  391 

attention  to  the  mixture,  and  that  particularly  in  the  cool  season 
a  stiff  mixture  is  used.  Steel  filings  and  a  small  percentage  of  car- 
borundum in  the  proportion  of  16  pounds  to  the  sack  of  cement 
produce  good  results. 

7.  Treatment  of  Floors.     A  concrete  floor  may  be  treated   in 
a  manner  somewhat  similar  to  a  wood  floor.     It  may  be  shellaced 
and  waxed  or  varnished  and  painted  if  desired.     Where  a  floor  has 
been  put  down  and  the  finish  is  unsatisfactory  from  the  standpoint 
of  dusting,  if  not  too  bad  the  trouble  may  be  remedied  by  a  coat  of 
floor  paint  made  with  a  thin  varnish  as  the  base.     The  thinner  the 
paint  of  the  first  coat  the  greater  its  penetration  and  the  better 
the  result  from  the  standpoint  of  reducing  the  tendency  to  dust. 

Where,  however,  the  surface  is  unusually  bad  there  is  no  remedy 
except  by  rubbing  it  down  with  the  carborundum  wheel  in  a  manner 
similar  to  that  in  which  the  finish  is  secured  in  terrazzo  floors. 

8.  Concrete  Stairs.     Reinforced  concrete  provides  an  inexpen- 
sive means  for  building  stairways  which  are  far  more  nearly  fire- 
proof than  any  other  type  which  can  be  constructed. 

The  accompanying  typical  detail  of  reinforcement  Fig.  95, 
shows  the  usual  method  of  reinforcing  employed  for  this  purpose. 

For  ordinary  runs,  such  as  twelve  feet,  f "  rounds,  6"  on  centers 
are  ample  for  the  inclined  slab.  The  inclined  slab  is  generally  built 
4J  or  5"  thick  for  ordinary  runs  and  the  horses  are  cast  on  top  of 
the  incline.  Where  fancy  treads  are  desired  they  are  sometimes  made 
of  white  Portland  cement  and  crushed  quartz.  This  makes  a 
very  durable  tread  and  a  material  which  in  its  good  appearance 
ranks  next  to  marble  and  will  wear  somewhat  better. 

When  marble  or  slate  treads  are  used  they  can  be  readily  bedded 
on  the  concrete  horse  and  the  riser  brushed  up,  rubbed  and  painted 
or  varnished  as  preferred.  Frequently  it  is  desirable  to  suspend 
stair  platform  supports  from  above.  This  can  very  readily  be 
done  by  dropping  the  slab  rods  down  to  the  level  of  the  platform 
from  the  slab  above  on  one  or  more  sides  and  encasing  the  suspender 
rods  in  an  ordinary  2"  partition  of  cement  plaster  for  fire  protection. 
Fastenings  for  metal  hand  rails  can  be  readily  cast  in  the  end  or  top 
of  the  stair  treads  as  the  work  is  placed. 

9.  Insulation   of   Roofs.     Those   not   familiar   with   reinforced 
concrete  frequently  make  the  mistake  of  designing  roof  slabs  in  a 
cold  climate  without  insulation.     The  result  is  that  the  moisture 
in  the  warm  air  in  the  room  below  the  roof  slab  is  condensed  on 


392 


CONCRETE    STAIRS 


the  underside  of  the  cold  slab  and  drips  continually  whenever  the 
slab  is  colder  than  the  air  within  the  room.  This  is  readily  remedied 
by  a  cinder  filling  from  four  to  six  inches  thick.  In  fact,  we  fre- 
quently recommend  to  our  clients  that  instead  of  putting  on  a  roof 
slab  proper  the  ceiling  slab  be  cast  level  which  may  at  some  future 
date  be  used  for  a  floor  should  a  story  be  added  and  on  this  slab  to 
build  up  with  cinders  sufficiently  to  give  the  standard  pitch-and-gravel 


Fig.  95.     Typical  Detail  of  Reinforcement  Concrete  Stairs 

roof  the  usual  slope.  This  slope  should  preferably  be  in  the  neigh- 
borhood of  5  /16  to  3/8  inch  to  a  foot  of  run.  On  the  top  of  the 
cinder  insulation  a  one  inch  coat  of  cement  mortar  mixed  about 
one  cement  to  three  sand  is  spread  which  forms  a  good  base  for  the 
pitch-and-gravel  roof.  In  place  of  the  cinder  filling  a  false  roof  is 
frequently  built  up  using  old  centering  lumber.  Where  this  is  done 
all  openings  thru  the  ceiling  and  roof  should  be  encased  or  protected 
by  a  concrete  fire  wall,  then  no  further  damage  can  occur  than  the 


PROTECTION    FOR    PLUMBING  393 

burning  up  of  the  comparatively  inexpensive  false  roof  should  the 
same  catch  fire  from  above.  The  ceiling  slab,  should  the  roof 
burn  away,  would  protect  the  goods  or  business  carried  on 
beneath  it  until  the  false  roof  could  be  replaced. 

Insulation  is  of  the  utmost  importance  in  concrete  roofs  in  the 
climate  of  the  United  States,  north  of  Southern  or  Central  Kansas. 
We  would  hesitate  to  allow  any  work  to  be  executed  with  our 
guarantee  against  this  difficulty  in  latitudes  north  of  this.  On  the 
Pacific  Coast  insulation  may  be  omitted  as  far  north  as  Portland 
without  serious  difficulty. 

10.  Protection    and    Provision    for    Plumbing.     In    plumbing 
fixtures  a  considerable  amount  of  lead  piping  is  used.     This  should 
be  either  entirely  eliminated  where  it  comes  in  contact  with  concrete 
or  well  protected  by  a  heavy  coating  of  tar  or  asphalt  paint  since  pieces 
of  lead  piping  uncoated  when  removed  from  concrete  are  often  found 
transformed  almost  completely  into  red  oxide. 

Cast  iron,  wrought  iron  or  steel  and  brass  fittings  are  not  in- 
juriously affected. 

In  general,  provision  should  be  made  in  the  pouring  for  all  large 
fixture  pipes,  since  digging  large  holes  in  the  concrete  should  not  be 
allowed  because  frequently  these  come  at  points  where  they  may 
seriously  weaken  the  construction. 

Cases  have  occurred  where  the  plumber  thotlessly  dug  a  hole 
right  thru  the  center  of  a  beam,  leaving  an  insignificant  amount  of 
concrete  each  side  of  the  hole  to  take  care  of  the  shear  and,  in  doing  so, 
cut  one  of  the  main  beam  rods,  thus  forcing  the  slab  to  carry  a  load 
provided  for  safely  only  by  the  portion  of  the  beam  cut  away. 

11.  Placing   Electric    Conduits,    Gas    Pipes,    etc.      The   most 
convenient  disposition  in  reinforced  concrete  work  of  pipes,  conduits, 
and  the  like,  is  to  bury  them  in  the  middle  of  the  slab  with  outlets 
at  desired  points.     So  buried,  conduits  if  of  moderate  size,  in  no 
wise  weaken  the  construction.     They  should  not,  however,  be  placed 
beneath  the  reinforcement.     This  is  a  mistake  that  is  too  often  made. 

Sometimes  conduit  pipes  are  placed  right  along  on  top  of  the 
centering  with  perhaps  f "  of  concrete  under  them  in  the  finished 
work,  and  reinforcing  bars  resting  on  top  of  the  conduit  pipe  dip 
downward  in  the  slab  on  each  side  of  the  conduit  to  a  greater  or  less 
extent.  Then,  as  soon  as  the  centering  is  struck  and  the  strain  comes 
upon  the  rods,  there  is  a  tendency  to  straighten  out  under  pull,  and 


394  PLACING    CONDUITS,    ETC. 

to  cause  the  slab  to  deflect  or  sag  and  open  up  large  unsightly  cracks 
near  the  bottom  of  the  conduit  pipe.  The  reduction  in  strength 
due  to  this  position  of  the  pipe  may  be  as  much  as  from  ten  to  twenty- 
five  percent  of  the  strength  of  the  slab.  In  the  ceiling  the  crack,  from 
the  standpoint  of  appearance  is  unsightly  and  leads  to  somewhat 
unwarranted  suspicion  of  extreme  weakness.  This  should  be  avoided. 

Standard  outlet  boxes  as  furnished  by  electric  supply  com- 
panies are  unfortunately  usually  much  too  shallow.  They  should 
be  deep  enough  so  that  the  pipe  connections  can  be  readily  made 
without  interference  with  the  reinforcement.  The  writer  has 
frequently  had  wood  plugs  turned  up  and  put  in  these  boxes  in  order 
to  keep  them  at  the  proper  elevation  and  give  an  opportunity  to 
place  the  conduits  without  bending  and  kinking  them  as  they  enter 
the  box. 

Provision  for  openings  in  floors  for  steam  pipes,  soil  pipes, 
leaders  and  the  like,  may  be  made  most  economically  by  placing 
thimbles  of  sheet  metal  (filled  with  sand)  on  the  forms  in  the  de- 
sired location,  thus  saving  the  expense  of  cutting  later. 

When  holes  have  to  be  cut  thru  the  slab  the  cutting  should  com- 
mence from  the  bottom.  If  the  hole  is  cut  thru  from  the  top,  as  soon 
as  the  drill  or  chisel  strikes  the  bars  a  large  unsightly  chunk  will 
be  broken  out  of  the  underside  of  the  slab.  Since  it  is  quite  difficult 
to  patch  these  places  with  plaster  the  architect  should  not  allow 
the  work  to  be  done  in  this  manner. 

12.  Plastering  on  Reinforced  Concrete  Work.  This  is  a  feature 
of  concrete  building  construction  which  is  of  considerable  interest 
to  the  architect.  It  is  decidedly  annoying  for  a  client  to  come  to 
the  architect  and  state  to  him  that  a  large  section  of  the  plaster 
has  dropped  off  from  certain  sections  of  his  building.  This  happens 
far  more  frequently  than  the  advocate  of  reinforced  concrete  likes 
to  admit,  altho  when  the  causes  of  the  failure  of  plaster  to  adhere 
to  the  work  are  fully  investigated  and  the  work  properly  executed 
there  is  little  trouble  from  this  cause. 

Plasterers  are  in  the  habit  of  plastering  on  wire  lath,  wood  lath 
or  the  like.  With  such  a  base  upon  which  to  work  there  is  ample 
opportunity  for  a  lean  mortar  to  clinch  in  a  firm  and  satisfactory 
manner.  When  plastering  on  concrete,  however,  plaster  is  held 
to  the  concrete  by  adhesion  only.  There  is  little  or  no  chance  for 
efficient  clinch  or  mechanical  bond  such  as  occurs  when  plastering 
on  lath  or  wire  cloth.  The  materials,  the  concrete  and  the  plaster, 


PLASTERING    ON    REINFORCED    CONCRETE    WORK  395 

which  do  not  have  exactly  the  same  coefficient  of  expansion  are  held 
together  by  adhesion  between  the  two,  and  evidently  this  will 
be  greater  the  richer  the  plaster  mortar.  It  will  be  greater  when 
the  surface  of  the  forms  used  for  centering  are  rough  sawed  lumber 
than  with  surfaced  lumber.  The  tendency  to  drop  off  will  be  less 
the  thinner  the  plaster  coat  and  less  damage  can  result  from  the 
fall  of  any  given  section  of  plaster;  hence,  a  thin  coat  of  plaster  is 
to  be  preferred  to  a  thick  scratch  coat  and  finish  coat  thereon. 

Lime  mortar  well  gaged  with  Portland  cement  just  before  use 
will  adhere  better  to  reinforced  concrete  than  the  gypsum  or  patent 
plasters.  Any  plaster  will  adhere  to  concrete  best  when  the  concrete 
is  thoroly  set  and  dry.  Trouble  almost  invariably  results  from  the 
attempt  to  plaster  on  concrete  before  it  has  had  a  chance  to  thoroly 
dry  out  and  set  hard,  as  it  seems  that  the  moisture  from  the  concrete 
prevents  the  plaster  from  drying  and  setting  properly. 

Washing  the  surface  of  the  concrete  before  plastering  with  a 
solution  of  one  part  vinegar  to  three  parts  clean  water  greatly 
improves  the  bond  between  the  two  materials,  since  it  removes 
the  inert  matter  from  the  surface  of  the  concrete. 

Some  plasterers  prefer  to  coat  the  concrete  work  with  R.  I.  W., 
or  other  tar  paint  in  advance  of  applying  the  plaster  in  order  to 
secure  a  more  satisfactory  bond. 

Considerable  trouble  has  occurred  with  plaster  upon  reinforced 
concrete,  tho  in  all  cases,  on  investigation,  it  has  been  found  either 
that  the  plaster  was  applied  upon  partially  cured  concrete  or 
improperly  put  on. 

Sometimes  the  plasterer  will  endeavor  to  put  on  a  thick  coat, 
get  air  bubbles  between  the  new  plaster  and  the  concrete  and  these 
expanding  and  contracting  with  each  change  of  temperature  will 
gradually  loosen  up  quite  a  large  area  of  the  plaster  coat  and  after 
six  or  eight  weeks  it  will  drop  off  in  large  chunks. 

The  remedy  for  this  difficulty  is  as  follows: 

First,  see  to  it  in  centering  the  floor  that  the  rough  side  of  the 
lumber  is  placed  next  to  the  concrete,  giving  a  rough  surface  rather 
than  a  smooth  surface  for  the  plaster  to  stick  to. 

Second,  see  to  it  that  the  concrete  work  is  thoroly  dried  before 
attempting  to  plaster  it. 

Third,  thoroly  wash  the  under  side  of  the  surface  of  the  slab 
with  the  vinegar  solution  recommended. 

Fourth,  see  to  it  that  a  rich  mortar  is  used. 


396  SUSPENDED    CEILINGS 

Fifth ,  make  the  finish  as  thin  as  possible,  a  skin  coat  1/16  to 
1  /32  inch  thick  being  ample  to  make  a  good  finish. 

Sixth,  avoid  the  use  of  soap,  grease  or  benzine  to  prevent  the 
concrete  from  adhering  to  the  centering. 

Nearly  all  of  the  patent  gypsum  plasters  will,  when  applied  wet 
to  steel  or  iron,  badly  corrode  the  metal.  Fortunately  this  corrosion 
seems  to  continue  only  until  the  plaster  sets  but  it  is  sufficient  to 
stain  the  plaster  badly  in  the  vicinity  of  the  metal.  It  may  be 
prevented  in  the  manner  recommended  for  the  protection  of  lead  in 
concrete. 

13.  Suspended  Ceilings.  Frequently  a  slab  is  put  up  where 
it  is  desired  to  suspend  a  ceiling  below,  either  to  conceal  pipes, 
flues  and  the  like,  or  as  insulation  for  the  roof.  This  is  readily 
arranged  in  the  following  manner: 

Take  ordinary  |"  round  wire,  make  a  3"  loop  on  the  upper  end 
and  drop  it  thru  a  hole  in  the  form.  It  will  then  be  anchored  in  the 
concrete  as  soon  as  the  concrete  is  cast,  and  the  free  end  may  be  used 
to  tie  up  angles,  tees  or  groove  irons  which  are  to  be  used  for  the 
ceiling  frame. 


397 


CHAPTER  XIV 

ARTISTIC  AND  COMMERCIALLY  PRACTICABLE  CONCRETE 
SURFACE  FINISHES. 

1.  Stipple  Coat.     It  is  found  that  by  applying  a  stipple  coat, 
either  rough  or  smooth  as  desired,  a  very  pleasing  effect  can  be  readily 
obtained.     An  example  of  this  treatment  is  shown  in  the  figure  of 
the  Smythe  block,  Wichita,  Kansas.     The  writer  has  adopted  this 
finish  for  most  of  his  bridge  work,  as  it  gives  a  greater  appearance 
of  strength,  readily  covers  up  the  minor  imperfections  in  the  cen- 
tering, and  affords  a  pleasing  contrast  with  the  highly  ornamental 
stone  railings  with  which  he  prefers  to  finish  his  work. 

The  stipple  coat  is  usually  applied  with  a  broom-corn  brush 
and  consists  of  a  thoroly  mixed  grout  of  a  neat  cement  one  part, 
and  sand  one  part.  Treatment  of  the  surface  should  be  as  follows: 

Wet  down  the  face  of  the  wall  thoroly  with  a  hose.  Then  apply 
the  stipple  coat,  spattering  it  on.  This  method  of  treatment  is 
being  adopted  to  a  large  extent  by  architects  in  the  finishing  of 
exterior  cement  plaster  walls  for  residences.  A  very  neat  effect 
indeed  is  obtained  in  this  manner  at  a  low  cost.  Expanded  metal 
wire  lath  is  nailed  to  the  studs,  plastered  with  a  Portland  cement 
mortar  mixed  usually  with  same  ten  percent,  of  hydrated  lime, 
the  mixture  being  practically  one  cement  to  one  and  one-half  sand 
and  finished  with  a  stipple  coat  as  outlined. 

Failure  of  cement  plastered  walls  may  be  attributed  in  the  main 
to  failure  of  the  contractor  to  use  enough  cement.  Too  many 
workmen  put  forward  the  argument  that  the  concrete  will  not 
be  good  if  it  is  made  too  rich,  while  as  a  matter  of  fact  it  requires 
a  rich,  strong  mixture  to  withstand  the  frost  and  severe  climate  in 
all  northern  states.  Plaster  work  which  would  stand  without  injury 
in  Cuba  and  Arizona  would  fall  to  pieces  in  short  order  in  Minnesota 
or  Manitoba.  A  properly  applied  cement  coat  of  a  rich  mortar, 
however,  will  stand  the  climatic  conditions  in  the  north  while  in  the 
south  also  a  rich  mixture  is  to  be  decidedly  preferred. 

2.  Plaster  Coat  on  Rough  Cast  Concrete.     A  very  expensive 
effort  was  made  to  secure  a  good  surface  finish  on  the  Grand  avenue 


398 


STIPPLE    FINISH 


viaduct  in  Milwaukee.     The  specifications  required  that  the  inside 
of  the  forms  be  lathed  with  expanded  metal  and  plastered  with  a 


Fig.  96.     Stipple  finish  of  Smythe  Block,  Wichita,  Kans.     Louis  Curtis,  Architect. 

mixture  of  plaster  of  Paris  and  lime.  This  plaster  coat  was  to  be 
oiled  in  advance  of  placing  the  concrete  and  the  concrete  to  be  placed 
and  tamped  in  layers.  On  removal  of  the  forms,  notwithstanding 
the  greatest  care  on  the  part  of  the  contractor,  the  line  of  demarka- 
tion  between  the  several  layers  was  plainly  visible  and  it  was  found 
impossible  to  put  up  the  work  without  blemish  as  required  in  the 
specifications. 


BRUSH    FINISH  399 

The  work  was  finally  carried  out  by  removing  the  forms  on  the 
exposed  surface  as  soon  as  practicable  and  plastering  with  a  thin 
coat  of  rich  cement  grout.  This  is  a  practice  which  is  not  to  be  rec- 
ommended because  wherever  the  mass  of  the  concrete  has  had  time 
to  get  fairly  hard  this  plaster  coat  is  liable  to  check  and  scale  off,  tho 
occasionally  it  has  been  applied  with  a  fair  degree  of  success  before 
the  concrete  has  had  time  to  thoroly  harden. 

3.  Finish  Obtained  by  Brushing  and  Washing.  Mr.  Henry  H. 
Quinby  of  Philadelphia  appears  to  have  been  one  of  the  first  to 
introduce  a  method  of  brushing  and  washing  the  concrete  surfaces, 
to  bring  into  relief  the  aggregate  used. 

The  process  consists  of  removing  the  forms  after  the  material 
has  set,  but  while  it  is  still  friable,  and  then  immediately  washing 
and  rinsing  the  cement  which  has  formed  against  the  mold  and 
thereby  expose  the  particles  of  sand  and  stone.  The  appearance 
then  depends  upon  the  character  of  the  aggregate  in  the  concrete 
as  respects  its  color  and  the  uniformity  of  its  distribution  in  the  mixture. 

The  time  to  be  allowed  for  setting  before  washing  must  depend 
upon  the  nature  of  the  cement  and  the  temperature  conditions. 
Quick  setting  cement  and  warm  weather  call  for  the  removal  of  the 
forms  in  from  seven  to  ten  hours.  The  appearance  may  be  con- 
trolled somewhat  by  the  extent  of  washing  which  may  be  to  the 
point  of  leaving  the  stone  aggregate  in  decided  relief  producing  a 
rough  coarse  texture  much  admired  by  most  architects. 

An  interesting  article  on  this  subject  will  be  found  in  a  book 
entitled  "  Concrete  Factories,"  by  Leslie,  published  by  Bruce  and 
Banning  of  New  York,  and  in  some  of  the  older  numbers  of  the 
'  'Cement  Age." 

A  well  written  paper  on  the  same  subject  has  been  published 
by  the  Universal  Portland  Cement  Company  in  their  trade  bulle- 
tins, numbers  54,  55  and  56,  which,  thru  the  courtesy  of  the  com- 
pany, is  reproduced  in  part  herewith  : 

The  ordinary  concrete  surface,  it  must  be  admitted,  is  any- 
thing but  pleasing  in  appearance,  being  usually  a  comparatively 
smooth,  lifeless  surface  of  a  somber  grayish  color.  It  makes  but 
little  difference  what  cement,  sand  or  aggregate  is  used,  or  in  what 
proportions  they  are  mixed,  the  general  aspect  of  the  unfinished 
form  surface  is  the  same.  There  may  be  the  greatest  difference  in 
color,  shade  and  texture  of  the  aggregate  used  in  two  separate 
concrete  surfaces,  yet  unless  they  are  so  treated  as  to  bring  out  and 
expose  the  aggregate,  the  resulting  surfaces  will  look  alike. 


400 


FINISHED    SURFACES 


Figure  II 


Figure  III 
(Surfaces  reduced  one-half  of  original.) 


FINISHED    SURFACES 


401 


Figure  IV. 


Figure  V. 


Reproductions  are  actual  size. 


402  AGGREGATE    OF    DIFFERENT    COLORS 

It  is  quite  difficult  to  distinguish  an  ordinary  unfinished  con- 
crete surface  in  which  bank  gravel  is  the  aggregate  from  one  in 
which  crushed  red  granite  is  used,  but  the  same  surfaces,  if  subjected 
to  any  one  of  a  number  of  different  methods  of  surface  treatment, 
will  present  a  marked  and  pleasing  contrast  in  appearance.  It  is 
the  monotonous  sameness  in  the  appearance  of  concrete  work  that 
architects  object  to  so  strongly.  To  show  what  can  be  accomplished 
in  producing  pleasing,  artistic  and  commercially  practicable  surface 
finishes  for  concrete  work  is  the  object  of  this  article. 

On  the  preceding  pages  are  found  photographic  reproductions  of 
brushed  concrete  surfaces.  The  difference  between  these  sur- 
faces and  that  of  ordinary  gravel  concrete  is  very  striking,  yet  they 
are  all  practical,  commercial  finishes,  and  can  be  obtained  by  the 
use  of  material  from  ordinary  gravel  banks. 

Figure  I  shows  a  comparatively  fine,  even-grained  surface, 
composed  of  one  part  Portland  cement  and  three  parts  of  fine  sand 
all  of  which  passed  a  No.  8  and  was  retained  upon  a  No.  50  mesh 
screen.  Figure  II  is  very  much  like  Figure  I  in  general  appearance 
and  color,  but  of  a  rougher,  more  uneven  texture.  This  surface  is 
a  1  :  mixture,  with  coarse  sand,  passing  thru  a  No.  4  and  retained 
on  a  No.  8  screen.  Figure  III  represents  a  finish  made  from  a  1  :  3 
mixture  of  cement,  and  J"  to  \"  pebbles.  Thus  these  surfaces  are 
identical  in  every  respect,  except  as  to  size  of  aggregate.  The  three 
surface  finishes  were  all  produced  by  the  same  method  of  treatment . 

The  cuts  give  but  a  poor  idea  of  the  appearance  of  the  actual 
surfaces,  as  the  color  and  texture  which  give  life  and  individuality  to 
the  surfaces  are  lacking.  To  appreciate  the  value  of  this  finish  for 
concrete  work,  the  surfaces  from  which  these  cuts  were  made  should 
be  seen. 

Figures  IV,  V  and  VI  are  three  cuts  from  photographic  reproduc- 
tions of  concrete  surfaces  similar  as  to  surface  treatment  to  those 
previously  shown,  but  differing  from  them  in  the  aggregates  used. 

Figure  IV  shows  a  decidedly  pleasing,  even-grained  surface  com- 
posed of  one  part  Portland  cement  and  two  and  one-half  parts  red 
granite  screenings,  all  of  which  passed  a  No.  8  and  was  retained  on 
a  No.  16  seive.  Figure  V  is  a  reproduction  of  a  surface  composed 
of  one  part  cement  to  two  and  one-half  parts  ordinary,  quarter 
inch,  granite  screenings,  the  material  passing  a  No.  8  sieve  being 
rejected.  Both  these  surfaces  are  quite  similar  in  every  respect 
in  texture,  that  represented  by  Figure  V  being  of  a  rougher 
texture  than  the  other.  As  the  cement  is  barely  perceptible  on  these 


BRUSH    FINISH    WITH    DIFFERENT    AGGREGATES  403 

surfaces  both  look  very  much  like  rough,  undressed  red  granite, 
the  color  being  practically  the  same  as  that  of  the  screenings  of 
which  they  were  made.  Figure  VI  represents  a  treated  surface 
composed  of  one  part  cement  to  two  and  one-half  parts  of  black 
pebbles,  varying  in  size  from  those  retained  on  a  No.  10  sieve  to 
those  passing  a  \\"  mesh.  The  cut  gives  but  a  poor  idea  of  the  pleas- 
ing contrast  between  the  light  colored  cement  background  and 
the  black  pebbles  which  stand  out  in  bold  relief  from  the  surface. 

Comparing  these  cuts  and  those  in  the  preceding  page,  quite  a 
variation  in  general  aspect  and  texture  is  to  be  noted,  and  an  exami- 
nation of  the  actual  surfaces  would  reveal  a  still  greater  difference 
in  appearance  owing  to  the  striking  variation  in  color  and  size  of  the 
aggregate  used.  Had  these  six  surfaces  been  left  untreated,  they 
would  have  looked  practically  alike. 

By  varying  the  kind,  size  and  proportions  of  the  aggregates, 
surface  finishes  of  practically  any  desired  color  and  texture  can 
be  obtained,  the  possibilities  being  limited  only  by  the  number  of 
different  aggregates  available  and  the  combinations  of  same.  A 
great  variety  of  finishes  may  be  produced  by  using  red  and  black 
granite  and  limestone  screenings,  black  and  white  marble  chips  and 
different  colored  pebbles  and  sands. 

All  the  cuts  shown  represent  brushed  concrete  surfaces,  the 
process  consisting  of  simply  brushing  the  surfaces  with  a  stiff  brush, 
permitting  it  to  harden  for  a  few  days  and  then  treating  it  with  a 
dilute  solution  of  hydrochloric  acid,  the  method  of  procedure  being 
as  follows: 

Having  decided  upon  the  general  color  scheme  and  texture  of 
the  desired  surface  the  first  step  is  the  making  and  treating  of  small 
sample  surfaces.  A  limited  amount  of  experimenting  with  the 
materials  available  will  always  prove  profitable.  The  color  and  tex- 
ture of  the  finished  surface  depends  upon  the  color,  size  and  pro- 
portions of  the  aggregates  used,  and  the  successful  reproduction  of 
the  desired  surface  is  dependent  upon  the  proper  selecting,  grading, 
proportioning  and  mixing  of  the  materials  and  the  proper  placing 
and  finishing  of  the  surface.  After  determining  by  experiment 
the  proper  size  and  proportions  of  aggregates  to  produce  the  desired 
effects  and  the  proper  consistency  of  the  mix,  adhere  strictly  to  them; 
that  is,  take  the  trouble  to  measure  the  materials  for  each  batch  of 
concrete  and  to  gauge  them  with  a  measured  amount  of  water. 
The  results  obtained  will  more  than  justify  the  extra  expense  this 
will  entail  over  the  all  too  prevalent  method  of  measuring  material 
by  wheelbarrow  loads  and  adding  the  water  with  a  hose;  in  fact, 


404  APPLYING    THE    FINISH 

uniform  results  cannot  be  obtained  unless  the  work  is  done  as 
pointed  out. 

The  slightest  imperfections  and  irregularities  in  form  surface 
are  transferred  to  the  concrete,  producing  unsightly  surfaces  when 
left  untreated,  and  a  pleasing  surface  cannot  be  obtained  by  a 
nicety  of  form  construction  alone.  For  brushed  surfaces,  all  that 
is  required  of  the  forms  is  that  the  face  lagging  be  kept  true  to  sur- 
face and  the  joints  be  tight.  For  surfaces  too  large  to  concrete  in 
one  day  the  forms  should  be  so  constructed  as  to  permit  of  the 
removal  of  sections  of  the  face  form.  This  can  be  accomplished 
by  setting  the  studs  or  uprights  back  a  few  inches  from  the  face 
lagging  and  connecting  both  by  means  of  cleats  and  wedges.  The 
face  forms  also  should  be  well  oiled  to  prevent  the  concrete  sticking 
to  the  forms.  In  large  areas  the  introduction  of  buttresses  and 
panels  or  the  breaking  up  of  the  surface  by  horizontal  joints  or 
courses  will  add  greatly  to  the  appearance,  the  joints  being  simply 
indentations  in  the  surface  produced  by  beveled  beads  fastened  to 
the  forms.  It  is  extremely  hard  to  join  two  different  days'  work 
so  that  the  joint  is  not  perceptible  and  unsightly,  and  the  breaking 
up  of  the  surface  as  indicated  will  greatly  assist  in  the  concreting 
if  care  be  taken  to  end  and  start  each  succeeding  day's  work  at  a 
course  or  joint. 

The  facing  material  should  be  from  one  to  one-and-a-half  inches 
thick,  the  remaining  thickness  of  the  work  being  composed  of  ordinary 
concrete,  but  the  facing  and  backing  must  be  deposited  at  the  same 
time  so  as  to  make  one  solid  mass,  thereby  insuring  perfect  bond. 
The  facing  material  may  be  applied  to  the  forms  just  ahead  of  the 
backing,  which  is  placed  against  and  rammed  into  it,  or  the  backing 
first  and  then  brushed  back  from  the  form  with  a  spade  and  the  fac- 
ing material  deposited  between  the  backing  and  the  form.  Both 
these  methods  have  been  successfully  used.  A  third  and  possibly 
the  best  method  of  placing  the  facing  material  consists  of  the  use  of 
what  might  be  called  a  metal  facing  form  or  mold,  constructed  and 
used  as  follows:  To  short  lengths  of  3/16"  iron  plates  8  or  10 
inches  wide  and  6  feet  long,  three  1  or  1J"  angles  are  riveted,  placing 
an  angle  at  the  center  of  the  plate  and  one  about  six  inches  from 
each  end.  One  edge  of  the  plate  should  be  slightly  flared  to  assist 
in  depositing  the  material  and  this  edge  provided  with  handles. 
The  metal  facing  plate  is  placed  against  the  wall  form  with  the 
handles  up  and  the  angles  tight  against  the  form.  The  space  between 
it  and  the  back  of  the  wall  filled  with  the  concrete  backing  and  the 
1  or  1J"  space  between  the  metal  form  and  the  face  form  filled  with 


BRUSHING    AND    CLEANING    THE    FINAL    SURFACE  405 

the  facing  material.     The  metal  form  is  drawn  almost  out,  and  after 
thoroly  tamping  the  backing  against  the  facing  the  process  is  repeated. 

For  brushed  surfaces  the  forms  must  be  removed  from  the  work 
as  soon  as  possible  and  the  concrete  surface  brushed  while  still 
green.  It  is  not  possible  to  state  how  old  the  work  should  be  before 
removing  the  forms  and  brushing  the  surface.  This  will  depend 
upon  a  number  of  conditions,  the  character  of  the  work,  cement 
and  aggregate  used,  consistency  of  the  mixture,  and  very  much  upon 
the  weather  conditions.  As  a  rule  in  hot  weather  the  forms  can  be 
removed  the  next  day  and  the  surface  brushed,  but  in  cold  weather 
the  facing  form  cannot  be  removed  so  soon,  several  days  perhaps  a 
week  being  required  for  the  concrete  to  attain  the  necessary  hard- 
ness and  strength.  Care  must  be  taken  that  the  brushing  is  not 
done  too  soon,  as  little  particles  of  aggregate  will  be  removed,  re- 
sulting in  a  pitted,  unsightly  surface.  On  the  other  hand  the  longer 
the  surface  stands  before  being  brushed  the  more  brushing  it  will 
require  to  remove  the  film  of  material  that  has  flushed  to  the  surface. 
Brushing  should  be  done  just  as  soon  as  it  can  be  without  removing 
particles  of  aggregate.  When  this  can  be  done,  can  only  be  deter- 
mined by  experimenting  with  the  particluar  surface.  An  ordinary 
scrubbing  brash  with  stiff  palmetto  fibers  or  a  metal  wire  brush 
will  answer  for  the  work.  Two  or  three  days  after  the  brushing 
the  surface  should  be  washed  down  with  a  dilute  solution  of  com- 
mercial hydrochloric  acid,  one  part  acid  to  two  or  three  parts  water. 
The  acid  should  be  applied  with  an  ordinary  calcimining  brush  and 
the  walls  thoroly  rubbed,  while  wet  with  the  acid,  with  a  stiff  vegetable 
fiber  brush.  The  acid  should  not  be  allowed  to  remain  on  the  sur- 
face for  any  length  of  time — not  over  half  an  hour —  and  should  be 
washed  off  with  a  hose  and  clean  water.  It  is  important  that  the 
surface  be  thoroly  washed  after  the  acid  treatment,  for  if  it  is  not 
it  will  have  a  mottled,  streaky  appearance. 

A  desirable  surface  can  be  obtained  by  simply  brushing  and 
then  washing  with  a  hose  and  clean  water,  but  the  final  acid  treat- 
ment in  connection  with  the  brushing  will  produce  a  still  better 
surface. 

This  method  of  treatment  removes  the  film  of  mortar  that  has 
flushed  to  the  surface,  exposes  the  aggregate,  erases  all  traces  of 
form  markings  and  produces  a  rougher,  more  artistic  surface.  The 
roughness  of  the  surface  breaks  up  the  light,  the  color  of  the  aggre- 
gate adds  variety  and  life,  and  we  have  a  pleasing,  artistic,  true 
concrete  surface. 


406 


TOOLING 


4.  Finish  by  Tooling.  Where  the  architect  is  not  limited  in 
point  of  cost,  an  excellent  effect  can  be  secuerd  by  tooling  the 
surface  of  the  concrete  either  by  hand  or  using  pneumatic  tools. 
The  effect  will  depend  largely  on  the  character  of  the  aggregate  and 


Fig.  97.     Cast  Stone  Railing,  bridge  at  Fergus  Falls,  Minn.     John  Lauritzen,  Contractor. 
C.  A.  P.  Turner,  Engineer. 

where  this  has  been  carefully  selected  the  finish  is  quite  attractive, 
especially  when  the  surface  is  broken  into  blocks  by  rustication  or 
grooves. 


CAST   STONE 


407 


The  expense  of  tooling  ranges  from  five  to  ten  cents  per  surface 
foot,  depending  on  the  equipment  used,  while  that  of  brushing  and 
washing  should  not  run  more  than  one-fifth  of  this  amount. 

5.  Cast  Stone.  Where  suitable  aggregate  is  available  an  ex- 
cellent building  material  is  made  by  casting  concrete  in  sand 
molds. 

The  process  is  similar  to  the  iron  molders'  art.  Wood  or  plaster 
patterns  are  used,  a  sand  mold  prepared  and  the  concrete  which 
is  to  be  cast  is  mixed  to  about  the  consistency  of  cream.  When  the 
resulting  material  has  been  tooled  it  is  hard  to  distinguish  it  from 
the  natural  stone. 


Fig.  98.     Ornamental  Cast  Stone   Railing. 

In  cost  it  cannot  be  manufactured  to  compete  with  the  natural 
stone  where  there  is  little  freight  to  pay,  but  where  the  work  is  at 
all  complicated  and  there  is  a  duplication  of  the  parts  and  quarries 
of  good  building  stone  are  not  situated  convenient  to  the  locality, 
there  is  a  good  field  for  this  product. 

It  has  been  very  successfully  manufactured  in  Toronto,  St. 
Louis,  and  other  parts  of  this  country  and  also  in  Germany. 

Success  in  cast  stone  work  depends,  first,  on  a  rich  mixture,  second 
upon  the  selection  of  the  proper  aggregate,  which  must  be  a  crushed 


408  CAST    STONE 

stone  or  hard  pebble  which  will  weather  without  disintegration, 
and  third,  on  the  proper  method  of  mixing  and  agitating  the  mixture 
until  it  is  desposited  in  the  sand  mould. 

The  mixture  must  be  semi-fluid  so  that  it  will  flow  and  fill  the 
mould  and  must  be  kept  continually  agitated  until  in  place  in  the 
mould.  Otherwise  separation  occurs  with  an  inferior  casting  as 
the  result. 

Details  of  the  process  of  manufacture  are  beyond  the  scope  of 
our  present  purpose.  The  preceding  statement,  however,  compar- 
ing the  mode  of  casting  to  the  iron  moulders  art  gives  a  clean  cut 
idea  as  to  the  method  pursued. 


409 


CHAPTER  XV. 

1.  The  Execution  of  Work.  Construction  work  of  any  kind 
involves  a  great  responsibility,  not  only  on  the  part  of  the  designer, 
but  also  on  the  part  of  those  in  charge  of  the  work,  and  that  re- 
sponsibility is  for  the  safety  of  those  erecting  the  work. 

Perhaps  the  erection  of  no  type  of  building  is  so  free  from 
hazard  and  risk  to  the  lives  of  those  erecting  it  as  reinforced  con- 
crete construction  when  scientifically  designed  and  intelligently 
executed. 

During  the  last  ten  or  twelve  years,  the  manufacturers  of  Port- 
land Cement,  have  through  improvements  in  methods  of  manu- 
facture and  great  reduction  in  cost,  placed  this  material  on  the 
market  at  such  reasonable  rates  that  it  has  given  a  remarkable 
impetus  to  the  construction  of  concrete  work  in  all  lines.  Since,  as  a 
material  of  construction,  it  has  but  recently  come  into  general  use, 
it  is  not  surprising  that  a  large  part  of  the  engineering  and  archi- 
tectural profession  have  not  yet  become  so  familiar  with  its  char- 
acteristics, but  that  designs  lacking  in  conservatism  from  a  scientific 
standpoint  have  been  frequently  made,  and  this  combined  with 
the  execution  of  the  work  by  unskilled  contractors,  has  resulted  in  a 
number  of  instances  in  needless  sacrifice  of  life  and  large  property 
losses,  such  as  a  more  thoro  knowledge  and  study  of  the  char- 
acteristics of  the  material  should  entirely  prevent. 

It  would  be  neglect  of  duty  to  fail  to  briefly  summarize  and 
to  call  attention  pointedly  to  those  properties  and  characteristics 
of  concrete  which  must  be  known  and  appreciated  by  the  engineer 
and  constructor  in  order  that  he  may  avoid  the  serious  disasters  into 
which  those  ignorant  or  forgetful  of  them  have  been  too  frequently 
led. 

The  Hardening  of  Concrete:  Concrete  may  be  defined  as 
an  artificial  conglomerate  stone  in  which  the  coarse  aggregate  or 
space-filler  is  held  together  by  the  cement  matrix.  The  cement 
should  conform  to  the  Standard  Specifications  for  Cement,  recom- 
mended by  the  American  Society  for  Testing  Materials.  * 

The  contractor  and  architect  should,  at  least,  see  to  it  that  the 
cement  is  finely  ground,  and  that  it  meets  the  requirements  of  the 

*Substantially  the  same  specifications  are  adopted  thruout  England  and  America. 


410  HARDENING  OF  CONCRETE 

boiling  test.  This  last  may  be  readily  made  by  forming  pats 
of  the  cement  of  3  J  to  4  inches  in  diameter  on  a  piece  of  glass,  knead- 
ing them  thoroly  with  just  enough  moisture  to  make  them  plastic, 
so  that  they  will  hold  their  shape  without  flowing,  and  taper  to  a 
thin  edge.  Store  the  pats  under  a  moist  cloth  at  a  temperature  of 
sixty-five  to  seventy-five  degrees  Fahr.  for  a  period  of  24  hours. 
Then  place  the  pats  in  a  kettle  or  pan  of  cold  water,  and  after  raising 
the  temperature  of  the  water  to  the  boiling  point,  continue  boiling 
for  a  period  of  four  hours.  If  the  pats  do  not  then  show  cracks, 
and  if  they  harden  without  cracking  or  disintegrating,  the  con- 
structor may  be  satisfied  that  the  cement  is  suitable  for  use  in  the 
work.  Coarse  grinding  reduces  the  sand-carrying  capacity  of  the 
cement,  and  its  consequent  efficiency. 

The  function  assigned  to  the  concrete  element  in  the  combina- 
tion of  reinforced  concrete  is  to  resist  compressive  stresses  in  bend- 
ing; but  when  first  mixed  the  concrete  is  nothing  more  than  mud, 
and  in  order  for  it  to  become  the  hard,  rigid  material  necessary  to 
fulfill  its  function  in  the  finished  work  it  must  evidently  pass  in  the 
process  of  hardening  thru  all  stages  and  varying  degrees  of  hardness 
from  mud  and  partly  cured  cement  to  the  final  stage  of  hard,  rigid 
material.  This  curing  or  hardening  being  a  chemical  process,  does 
not  occur  in  any  fixed  period  of  time,  save  and  except  the  temper- 
ature conditions  are  absolutely  constant.  Hence  the  time  at  which 
forms  may  be  safely  removed  is  not  to  be  reckoned  by  a  given  number 
of  days,  but  rather  it  must  be  determined  by  the  degree  of  hardness 
attained  by  the  cement.  In  other  words,  during  warm  summer 
weather,  concrete  may  become  reasonably  well  cured  in  twelve  or 
fifteen  days.  If  the  weather,  however,  is  rainy  and  chilly,  it  may 
not  become  cured  in  a  month.  In  the  cold,  frosty  weather  of  the 
spring  and  autumn,  unless  warm  water  is  used  in  the  mix,  the  con- 
crete may  require  two  or  three  months  to  become  thoroly  cured, 
while  by  heating  the  mixing  water,  whenever  the  temperature  is 
below  50  degrees  Fahr.,  the  concrete  will  harden  approximately  as 
it  does  during  the  more  favorable  season. 

Concrete  which  has  been  chilled  by  the  use  of  ice  cold  water, 
or  that  has  become  chilled  within  the  first  day  or  two  of  the  time  it 
is  cast,  has  this  peculiarity,  that  it  is  very  difficult  indeed  for  the 
most  expert  to  determine  when  it  is  in  such  condition  that  it  will 
retain  its  shape  after  the  removal  of  the  forms.  Once  having  been 
chilled  in  the  early  stages,  it  goes  thru  successive  stages  of 
sweating  with  temperature  changes,  and  during  these  periods  it 
sometimes  happens  that  the  concrete  diminishes  in  compressive 


POURING    CONCRETE  411 

strength,  and  if  the  props  are  removed  it  sags  and  gets  out  of  shape. 
Such  deformation  will  generally  result  in  checks  and  fine  cracks, 
though  there  may  not  be  any  serious  diminution  of  the  ultimate 
strength.  These  checks  may  be  prevented  as  explained  above  by 
the  simple  method  of  heating  the  mixing  water  whenever  the  tem- 
perature has  dropped  below  50  degrees  Fahr.  In  colder  weather, 
that  is  below  the  freezing  point,  not  only  must  the  water  be  heated, 
but  as  a  rule  the  sand  and  stone  too,  also  a  little  salt  may  be  ad- 
vantageously used,  as  discussed  in  Chapter  I,  Section  15,  Page  30. 
The  work  must  then  be  properly  housed  and  kept  warm  for  at 
least  three  weeks  subsequent  to  pouring. 

Pouring  Concrete.  Bad  work  frequently  results  from  improper 
pouring,  or  casting  of  the  work.  In  filling  the  forms,  the  lowest 
portion  of  the  forms  must  be  filled  first.  A  column  should  be 
filled  from  the  center  and  not  from  the  side  of  the  cap.  Filling 
from  the  center  will  insure  a  clean  smooth  face  when  the  forms  are 
removed.  Filling  from  the  side  will  frequently  give  a  bad  surface 
because  the  mortar  will  flow  into  the  center  of  the  column  through 
the  hooping,  leaving  the  coarse  aggregate  with  voids  unfilled  at  the 
outside.  As  more  concrete  is  then  poured  in,  the  voids  between  the 
core  and  the  out-side  portion  will  become  filled,  and  the  soft  mor- 
tar will  not  be  able  to  flow  back  to  completely  fill  the  voids  between 
the  hooping  and  the  casing.  Where  the  spacing  of  the  hooping  is 
wide,  this  is  not  so  important,  but  it  becomes  very  important  where 
the  spiral  used  has  close  spacing.  It  is  better  to  cast  the  column  and 
mushroom  frame  complete,  continuing  to  pour  the  concrete  over 
the  center  of  the  column  so  that  it  always  flows  from  the  column 
into  the  Mushroom  slab  rather  than  the  reverse.  All  splices  must 
be  made  in  a  vertical  plane,  in  a  beam  preferably  at  the  middle  of 
the  span,  and  in  a  slab  at  a  center  line  of  a  panel. 

Separation  of  Materials.  In  pouring  concrete  where  the  mix  is 
too  sloppy,  separation  of  the  material  is  liable  to  occur.  This  is 
particularly  the  case  in  filling  columns  where  with  too  sloppy  a  mix 
layers  of  sand  and  gravel  and  cement  may  result  instead  of  a  con- 
crete of  uniform  composition. 

In  spouting  concrete,  careful  attention  should  be  given  to  the 
mixture  at  the  point  of  discharge,  since  if  the  inclination  of  the 
spout  is  too  great,  considerable  separation  occurs  and  inferior  con- 
crete is  the  result,  and  where  such  separation  occurs  the  concrete 
should  be  re-mixed  before  allowing  it  to  be  deposited  in  its  final 
position  in  the  work. 


412  TEST    FOR    HARDNESS.       LAP 

Test  of  Hardness  in  Warm  Weather.  We  have  pointed  out 
that  the  criterion  governing  the  safe  removal  of  forms  is  the  hard- 
ness or  rigidity  of  the  concrete.  A  test  of  hardness  in  concrete 
not  frozen  may  be  made  by  driving  a  common  eight-penny  nail 
into  it;  the  nail  should  double  up  before  penetrating  more  than 
half  an  inch.  The  concrete  should  further  be  hard  enough  to 
break  like  stone  in  knocking  off  a  piece  with  the  hammer. 
Noting  the  indentation  under  a  blow  with  the  hammer,  gives  a 
fair  idea  of  its  condition  to  those  having  experience. 

Sub-centering  is  a  desirable  method  of  preventing  deformation, 
where  the  use  of  the  forms  is  desired  for  upper  stories  before  the 
concrete  is  fully  cured. 

Test  for  Hardness  in  Cold  Weather.  Concrete  freshly  mixed 
and  frozen  hard  will  not  only  sustain  itself  but  carry  a  large 
load  in  addition,  until  it  thaws  out  and  softens,  when  collapse  in 
whole  or  in  part  is  inevitable.  Partly  cured  concrete  if  frozen, 
sweats  and  softens  with  a  rise  in  temperature,  hence  in  cold  weather 
there  is  danger  of  mistaking  partly  cured  concrete  made  rigid  by 
frost  for  thoroly  cured  'material.  In  fact  the  only  test  that  can 
be  depended  upon  with  certainty  in  cold,  frosty  weather,  is  to  dig  out 
a  piece  of  concrete,  place  a  sample  on  a  stove  or  hot  radiator,  and 
note  whether,  as  the  frost  is  thawed  out  of  it,  it  sweats  and  softens. 
This  gives  the  builder  and  engineer  a  perfectly  conclusive  test  of  the 
condition  of  the  concrete  as  to  whether  it  is  cured  or  merely  stiffened 
up  by  frost. 

Lap  of  Reinforcement  over  Supports.  Thoroly  tying  the  work 
together  by  ample  lap  of  the  reinforcement  is  a  prime  requisite 
for  safety  in  any  form  or  type  of  construction.  This  general  pre- 
caution insures  toughness,  and  prevents  instantaneous  collapse, 
should  the  workmen  exercise  bad  judgment  in  prematurely  remov- 
ing forms. 

2.  Responsibility  of  the  Engineer.  The  steps  which  it  is  possible 
for  the  engineer  to  take  in  securing  a  safe  construction  are  limited 
in  the  first  place  to  the  production  of  a  conservative  design,  and  one 
which  will  present  toughness,  so  that  its  failure  under  overload 
or  under  premature  removal  of  the  forms  will  be  slow  and  gradual 
This  he  can  do,  and  this  it  is  believed  he  is  morally  bound  to  do.  On 
the  other  hand,  he  cannot  design  reinforced  concrete  work  which 
will  hold  its  shape  without  permanent  deformation,  unless  it  is 
properly  supported  until  the  concrete  has  had  time  under  proper 
conditions  to  become  thoroly  cured. 


DANGEROUS    DETAILS 


413 


The  engineer  is  accountable  for  the  selection  of  a  type  of  design 
which  is  safe  to  erect.  That  is,  a  design  in  which  a  sudden  collapse 
cannot  readily  occur.  He  should  so  design  his  work  that  it  can  be 
executed  by  the  exercise  of  ordinary  care.  He  should  design  it  so 
that  there  shall  be  a  minimum  chance  of  bad  work  or  disastrous 
results  thru  lack  of  care  on  the  part  of  workmen. 

We  have  called  attention  to  the  fact  that  concrete  is  a  material 
naturally  best  fitted  for  monolithic  construction,  that  the  natural 
concrete  types  are  best  tied  together  by  so  reinforcing  the  construc- 
tion that  it  will  act  as  a  continuous  monolith.  To  do  this  ample 
lap  of  the  bars  is  essential  over  all  supports  whether  bearings  or 
supporting  columns. 

Every  failure  in  concrete  construction  is  detrimental  to  all  who 
are  engaged  in  this  line  of  business,  regardless  of  the  system,  type 
of  construction,  or  particular  reason  for  the  collapse,  and  accord- 
ingly all  engaged  in  this  line  have  a  like  interest  in  tracing  out  the 
cause  and  profiting  by  the  lesson  of  every  mishap  which  occurs. 


Fig.  99.  Detail  of  beam  causing  trouble  through  insufficient  lap  of  reinforcement  over  support. 

The  accompanying  detail  shows  the  beam  and  slab  reinforce- 
ment of  a  structure  which  collapsed  during  erection  and  the  char- 
acteristics of  this  failure  are  worthy  of  note,  as  due  to  insufficient 
laps  over  the  supports.  This  failure  started  in  an  upper  story 
where  the  small  diameter  of  the  column  gave  little  or  no  lap  of  the 
steel  over  the  supporting  concrete  while  the  lower  stories  where  the 
columns  provided  greater  overlap  were  erected  without  mishap 
until  broken  by  the  fall  of  the  upper  stories.  Where  there  is  in- 
sufficient lap,  owing  to  the  shrinkage  of  the  concrete  in  setting,  we 
have  not  only  the  shear  on  partly  hardened  concrete  but  also  tensile 
shrinkage  stresses  tending  to  crack  the  concrete  thru  at  the  point 
where  the  bars  are  not  sufficiently  lapped  over  the  supports. 

A  further  weakness  in  the  detail  illustrated  lies  in  the  fact  that 
none  of  the  reinforcement  for  positive  moment  was  carried  up  over 
the  support  to  resist  negative  moment  as  in  the  Hennebique  and 
Turner  types  of  continuous  beams,  illustrated  in  Chapter  III. 
This  detail,  viz.  the  carrying  of  a  portion  of  the  reinforcement 


414  RESPONSIBILITY    OF    THE    DESIGNER 

in  a  continuous  bar  over  and  beyond  the  support,  enables  a  portion 
of  the  load  to  be  carried  by  such  bars  as  in  a  swing  and  greatly 
relieves  the  shear  stress  from  such  loads  as  may  be  brought  upon 
partly  hardened  concrete,  and  thus  increases  the  safety  of  the  work 
during  the  critical  period  of  construction,  or  renders  failure  slow 
and  gradual  under  these  conditions  instead  of  sudden  and  without 
warning,  should  the  partly  cured  work  be  over-loaded.  This  safe- 
guard it  is  within  the  province  of  the  engineer  to  provide.  No 
excuse  can  be  made  for  failure  so  to  do  on  the  ground  of  increased 
cost  or  special  patent  monopoly  standing  in  the  way. 

Shearing  and  tensile  resistance,  as  has  been  noted,  is  developed 
during  curing,  at  a  less  rapid  rate  than  compressive  resistance  and 
hence  any  reasonable  safe-guard  of  value  such  as  that  just 
described  should  not  be  neglected  by  the  engineer. 

It  has  been  almost  invariably  the  case  where  combinations  of 
tile  and  concrete  have  been  used,  that  a  failure  starting  on  one  floor 
has  carried  with  it  one  floor  after  another  to  the  basement.  Such 
failures  do  not  occur  in  well  designed  reinforced  concrete  struc- 
tures. Hence  where  economic  conditions  permit,  the  engineer  is 
accountable  to  a  large  extent  for  the  selection  of  the  safer,  tougher 
types  of  construction  in  place  of  a  fragile  construction  which  may  be 
readily  destroyed  by  the  impact  of  any  large  mass  accidentally 
falling  upon  it. 

In  most  cases  where  failures  have  occurred,  had  the  centering 
been  left  in  place  for  a  period  of  from  four  to  six  months,  in  the 
authors'  judgment,  the  work  would  have  stood  and  no  serious  trouble 
would  have  resulted.  On  the  other  hand,  they  are  unable  to  regard 
that  kind  of  design  as  legitimate  which  must  necessarily  be  treated 
with  this  extreme  degree  of  care,  there  being  no  excuse  for  designing 
in  a  manner  which  leaves  an  opportunity  for  sudden  and  complete 
failure  of  the  work. 

The  engineer  designer  is  responsible  for  failure  to  provide  a  type 
of  column  design  in  which  there  are  no  obstructions  in  the  shaft 
of  the  column  to  interfere  with  securing  a  solid  casting.  Column 
failure  in  a  number  of  structures  under  construction  have  occurred 
where  the  longitudinal  column  reinforcement  was  arranged  with 
prongs  projecting  into  the  body  of  the  column  but  with  no  ties 
binding  the  longitudinal  steel  together.  These  prongs  interfered 
with  the  flow  of  the  concrete  material  in  the  shaft  of  the  column, 
interrupt  the  concrete  in  its  descent,  leaving  large  voids  which  in 
some  cases  caused  failure  when  the  forms  are  removed.  The  use 
of  this  type  has  fortunately  to  a  large  extent  been  discontinued. 


GREATER    SAFETY    WITH    NATURAL    CONCRETE    TYPES  415 

In  the  case  of  an  eleven  story  building,  some  years  ago,  the 
designer  used  vertical  reinforcing  bars  in  the  columns  and  tied  them 
across  the  shaft  with  numerous  quarter  inch  ties.  In  pouring  the 
concrete  in  several  columns  these  ties  blocked  the  flow  of  the  con- 
crete and  when  the  forms  were  removed  large  voids  were  found 
two  or  three  feet  in  length  in  several  columns  thru  the  interference 
of  the  ties  in  pouring  the  column.  There  is  no  excuse  for  the  em- 
ployment of  such  details. 

A  slab  reinforced  in  two  directions  and  supported  on  four  sides 
may  be  loaded  until  it  is  cracked  thru  and  if  the  slab  is  a  large  one 
may  be  loaded  until  the  deflection  is  twelve  or  fifteen  inches  and 
still  carry  the  load  which  broke  the  construction  down  at  this  point 
and  strained  the  steel  beyond  the  yield  point  value. 

A  slab  reinforced  in  one  direction  only  will  on  the  other  hand 
break  down  completely  and  sometimes  let  go  quickly  and  almost 
without  warning.  This  is  especially  true  where  forms  are  prema- 
turely removed. 

We  have  noted  in  Chapter  X  under  "  Elements  of  Economic 
Construction,"  that  the  true  concrete  types  which  are  continuous 
monolithic  construction  have  the  lowest  coefficient  of  bending,  hence 
there  is  little  excuse  on  the  part  of  the  designer  for  failure  to  adopt 
the  safest  type  of  construction,  particularly  in  view  of  the  fact  that 
it  may  be  figured  with  greater  certainty  and  a  higher  degree  of 
scientific  accuracy  than  the  types  of  simple  beam  or  one  way  rein- 
forcement that  have  been  used  in  this  composite  type  of  structure. 

While  the  engineer  may  be  held  responsible  for  accurate  com- 
putation and  for  features  that  bear  upon  the  safety  of  the  design, 
he  cannot,  unless  on  the  ground,  prevent  the  inexperienced  foreman 
from  knocking  centers  at  too  early  a  period.  He  cannot  prevent 
the  deflection  of  reinforced  concrete  work  where  the  material  has 
not  been  allowed  sufficient  time  to  properly  harden  before  the  re- 
moval of  the  forms.  If,  however,  his  design  is  one  of  the  two 
natural  concrete  types  of  construction  there  is  little  danger  of  a  sudden 
collapse  and  the  worst  that  can  happen  will  probably  be  the  neces- 
sity or  digging  out  and  replacing  some  work  which  has  got  out  of 
shape  owing  to  lack  of  judgment  and  haste  on  the  part  of  the  erec- 
tion superintendent. 

The  superintendent  on  the  job  should  make  a  special  point  of 
inspection  of  all  points  of  the  construction  where  cantilever  action 
is  depended  upon  for  stiffness  and  strength.  Wherever  such  action 


416  RESPONSIBILITY    OF    SUPERINTENDENT    OF    CONSTRUCTION 

is  required,  the  steel  should  be  at  the  top  and  its  position  at  the 
top  should  be  made  certain  by  such  inspection.  Far  too  frequently 
carelessness  has  been  exhibited  in  this  respect  and  unsatisfactory 
results  secured  from  the  standpoint  of  strength  and  service. 

3.  Responsibility  of  the  Constructor  and  Engineer  Superinten= 
dent.     The  constructor  is  primarily  responsible — 

For  the  honest  execution  of  the  work. 

For  the  proper  housing  of  the  cement  and  the  simple  methods 
of  determining  the  fineness  and  quality  as  recommended  in  Chapter  I. 

He  is  resposible  for  the  use  of 'sufficient  cement. 

Securing  a  proper  aggregate  and  seeing  that  the  mixture  of 
concrete  has  the  proper  consistency  to  produce  good  work,  i.  e., 
that  the  stone  is  of  proper  hardness,  suitable  size  and  free  from 
dirt  and  mud.  That  reasonably  clean,  coarse  sand  is  used. 

He  should  inspect  the  centering  and  see  that  it  is  erected  of 
proper  strength  and  that  ledgers  and  posts  are  properly  braced  so 
that  collapse  cannot  occur  during  erection. 

He  should  be  responsible  for  the  exercise  of  care  in  leaving  the 
forms  in  place  until  the  concrete  has  become  properly  cured. 

He  should  see  that  splices  are  properly  made  between  old  and 
new  work  and  that  segregation  and  separation  of  the  concrete  does 
not  occur  in  pouring. 

He  should  see  that  the  reinforcement  is  placed  as  required  by 
the  engineer's  plans  and  while  he  cannot  be  held  responsible  for  the 
design  he  should  exercise  greater  care  in  putting  up  the  less  con- 
servative types  which  consist  of  one-way  reinforcement  than  is 
essential  in  putting  up  multiple  way  systems  or  natural  concrete 
types. 

He  can  determine  whether  the  steel  furnished  is  of  reasonable 
quality  by  the  bending  test  and  by  nicking  and  breaking  so  that 
the  fact  is  ascertained  whether  the  metal  is  of  the  undesirable 
material  known  as  bushel  steel  or  a  uniform  quality  of  good  metal. 
This  is  of  special  importance  in  tensile  reinforcement  in  slab  and 
beam  steel,  hooping  and  the  like,  and  of  less  consequence  in  rein- 
forcement for  compression. 

4.  Significance  of  Cracks   in   Reinforced   Concrete.     Concrete 
in  setting  shrinks,  and  sometimes  cracks  by  reason  of  this  shrinkage, 
particularly  when  it  hardens  rapidly,   as  it  does  in  hot  weather. 


SIGNIFICANCE    OF    CRACKS    IN    CONCRETE    WORK  417 

This  shrinkage  sets  up  certain  stresses  in  the  concrete,  which,  com- 
bined with  temperature  changes,  occasionally  manifest  themselves 
by  subsequent  cracks  in  the  work.  Such  checks  or  cracks  do  not 
of  necessity  indicate  weakness,  providing  the  concrete  is  hard  and 
rigid,  since  the  steel  is  intended  to  take  the  tensile  stresses  and  the 
concrete  the  compressive.  Such  checks  sometimes  cause  an  un- 
warranted lack  of  confidence  in  the  safety  and  stability  of  the  work 
arising  from  the  common  lack  of  familiarity  with  the  character- 
istics of  the  material.  For  example,  the  owner  of  a  frame  building 
would  never  imagine  it  to  be  unsafe  because  he  found  a  few  season 
checks  in  the  timber.  He  is  sufficiently  familiar  with  the  seasoning 
of  timber  to  understand  how  these  checks  occur,  and  that  in  most 
instances  they  do  not  mean  a  loss  of  strength,  since,  as  the  timber 
hardens  by  thoroly  drying  out,  it  becomes  stronger,  as  a  rule,  to 
an  amount  in  excess  of  any  slight  weakness  which  might  be  developed 
by  ordinary  season  cracks  or  checks.  So  in  concrete,  when  the 
general  public  becomes  more  familiar  with  its  characteristics  they 
will  regard  as  far  less  important  than  they  now  do,  checks  which 
are  produced  by  temperature  and  shrinkage  stresses,  or  possibly 
by  slight  unequal  settlement  of  supports. 

Like  timber,  concrete  grows  harder  and  stronger  with 
time,  so  that  the  ordinary  temperature  check  does  not  reduce  the 
strength  of  the  work  as  much  as  the  hardening  of  the  concrete  with 
age  increases  it  as  the  steel  is  the  tensile  element  and  the  compres- 
sive element,  the  concrete,  having  grown  stronger  with  time,  the 
strength  of  the  combination  has  usually  increased  more  than  the 
decrease  in  strength  brought  about  by  the  check. 

Taking  the  modulus  of  elasticity  of  the  concrete  at  2,000,000, 
and  the  tensile  strength  of  concrete  at  300  pounds  per  square  inch, 
with  coefficients  of  expansion  of  .0000065,  if  the  ends  of  a  slab  or 
beam  are  rigidly  fixed  it  would  require  a  drop  of  24  degrees  below 
the  mean  temperature  at  which  the  concrete  hardens  to  stress  the 
concrete  in  tension  up  to  its  ultimate  capacity.  The  ends  of  our 
beams  and  slabs  are  rarely  absolutely  fixed,  as  the  walls  can  generally 
go  and  come  slightly  and  accommodate  some  temperature  change. 
A  certain  amount  of  temperature  reinforcement,  however,  should 
always  be  provided  where  the  reinforcement  is  in  one  direction 
only.  Eight  hundredths  of  one  percent  should  be  the  minimum  in 
slabs.  Even  with  this  reinforcement  or  with  multiple  way  rein- 
forcement, shrinkage  combined  with  temperature  will  occasionally 
cause  cracks  in  the  work.  The  season  of  the  year  and  the  tempera- 


418  PATENTS    ON    REINFORCED    CONCRETE 

ture  conditions  at  which  the  work  is  cast  or  the  condition  of  the 
cement,  all  play  their  part  in  producing  this  phenomenon  and  while 
the  constructor  can  guarantee  safe  work  from  the  standpoint  of 
strength,  he  cannot  guarantee  with  certainty  that  temperature 
cracks  will  not  occur. 

Their  occurrence  is  less  frequent  by  far  with  the  natural  types  of 
concrete  and  multiple-way  reinforcement  than  with  one-way  slab 
and  girder  construction,  such  as  Type  II,  or  the  combination  of 
tile  and  concrete  discussed  elsewhere. 

Where  structural  steel  frame  work  is  fireproofed  with  concrete, 
or  concrete  slabs  are  built  in  or  supported  by  and  made  integral 
with  structural  shapes  and  beams,  temperature  cracks  are  much 
larger  and  more  noticeable  than  with  true  reinforced  concrete  types 
for  the  reason  that  altho  the  coefficient  of  expansion  of  concrete  and 
steel  is  substantially  the  same,  the  specific  heat  and  conductivity 
of  the  two  materials  is  widely  different.  Hence  where  the  steel  is 
placed  in  the  concrete  in  large  sections  it  responds  more  quickly 
to  changes  in  temperature  than  does  the  concrete  envelop  and  ac- 
cordingly large  checks  and  cracks  are  a  common  occurrence.  Further, 
the  large  section  of  steel  separates  the  concrete  surrounding  it, 
causing  a  weakness  which  is  manifest  by  the  concentration  of  the 
temperature  effect  at  the  weak  section,  accounting  for  the  results 
noted. 

5.  Encouragement  to  Progress  in  the  Concrete  Industry  by 
Patents.  The  Federal  Constitution,  Art.  I,  Section  8,  provides 
that  Congress  shall  have  the  power  to  encourage  the  progress  of 
science  and  promote  the  useful  arts  by  securing  to  authors  and 
inventors,  for  limited  periods,  exclusive  rights  to  their  inventions 
and  discoveries.  In  accordance  with  this  authorization  and  in 
persuance  of  the  object  mentioned,  Congress,  by  the  enactment 
of  Section  4886,  Revised  Statutes  of  the  United  States,  provides 
that  any  person  who  has  invented  or  discovered  any  new  and  use- 
ful art,  machine,  manufacture  or  composition  of  matter,  or  any 
improvement  thereof,  may  obtain  a  patent  therefor  under  certain 
prescribed  rules  and  restrictions. 

Statute  law  identical  with  this  has  been  in  force  since  April 
10,  1790,  except  that  the  conditions  and  limitations  relating  to  it 
have  been  modified  somewhat  from  time  to  time. 

The  worker  in  concrete-steel  construction  is  naturally  interested 
in  coming  to  a  complete  understanding  of  the  scope  and  the  extent 
of  protection  afforded  him  by  this  statute. 


STATUTE  LAW  PROVIDING  PATENT  PROTECTION  419 

*The  word  " discovery"  is  not  used  either  in  the  Constitution 
or  the  Statute,  with  its  broadest  significance.  In  these  documents 
it  is  a  synonym  for  the  word  " invention,"  and  in  them  it  means 
nothing  else.  The  discoveries  of  inventors  are  inventions.  The 
same  man  may  invent  a  machine  and  may  discover  a  law  of  nature. 
For  doing  the  first  of  these  things  the  patent  laws  may  reward  him 
because  in  so  doing  he  is  an  inventor,  but  under  those  laws  he  can- 
not be  rewarded  for  discovering  a  law  of  nature  because  he  has 
originated  or  invented  nothing  by  this  act. 

A  discovery,  or  the  devising  of  some  means  to  utilize  a  discovered 
law  of  nature  in  a  new  and  novel  manner  is  an  invention. 

tThe  Statute  provides  that  a  grant  of  a  patent  may  be  made, 
and  it  says  that  the  grant  shall  be  limited  in  three  respects: 

1.  For  respective  discoveries,  and  hence  to  the  inventor  and  no 
one  else. 

2.  For  limited  times,  and  hence  no  perpetual  monopoly. 

3.  For  useful  art,  and  hence  every  patent  must  possess  utility. 
The  character  of  inventions  are  broadly  divided  into  six  classes: 
1.     A  machine.     2.  A  manufacture.     3.  A  composition  of  mat- 
ter.    4.  An   art.     5.  An   improvement    in   a   machine    or   process. 
6.  A  design. 

Those  engaged  in  the  industry  of  reinforced  concrete  construc- 
tion are  not  interested  from  the  standpoint  of  patent  protection  in 
reinforced  concrete  as  an  art,  for  as  such  the  embedment  of  iron  or 
metal  in  a  concrete  matrix  has  been  practiced,  as  we  have  pointed 
out  in  our  historical  sketch,  since  the  time  of  the  Roman  Empire, 
and  in  modern  times  to  a  considerable  extent  since  1850.  As  used 
in  a  building  or  bridge,  a  patent  for  a  design  would  offer  little 
protection. 

Reinforced  concrete  cannot  be  logically  classified  as  a  composi- 
tion of  matter  for  there  would  be  no  means  of  distinguishing  be- 
tween different  concrete  designs  from  this  standpoint,  as  having 
different  degrees  of  utility  and  strength.  If  it  is  treated  from  the 
standpoint  of  a  manufacture,  the  same  process,  the  same  machine, 
the  same  tools  and  the  same  general  classes  of  material  are  used 
in  the  manufacture  of  all  concrete  structures.  The  mere  form  of  a 
building  and  shape  of  a  room  is  not  patentable  if  the  decision  of  the 
Court  of  Appeals  in  the  Folding  Bed  Company  case  is  considered 

*See  Walker  on  Patents,  Art.  1,  Chap.  2. 
fMacomber,  Hand  Book  of  Patents. 


420  CLASSIFICATION    OF   KEINFORCED    CONCRETE    STRUCTURES 

conclusive.     Hence  the  mere  external  form  of  a  structure  cannot 
form  the  basis  of  a  valid  claim. 

If  we  treat  the  reinforced  concrete  structure  from  the  stand- 
point of  its  mode  of  operation  as  a  mechanism,  we  have  here  a  means 
of  differentiating  between  the  mechanical  efficiency  and  operation 
of  different  arrangements  of  reinforcement  in  the  same  matrix, 
which  results  in  differences  in  the  strength  and  stiffness  of  the  struc- 
tures using  Ihe  same  quantity  of  metal  and  concrete.  Hence  im- 
provement in  the  design  of  reinforced  concrete  structures  can  be 
rewarded  by  our  patent  laws  only  as  viewed  in  the  light  of  an  im- 
provement in  their  mode  of  operation  which  enables  us  to  differ- 
entiate one  type  or  genus  from  another  and  to  differentiate  between 
different  forms  of  the  same  species.  It  is  on  this  theory  that  the 
fixed  practice  of  the  United  States  Patent  Office  is  founded  in  grant- 
ing patents  on  the  different  types  of  design  of  reinforced  concrete 
members  and  structures. 

This  statement  is  substantiated  by  the  many  decisions  of  the 
Primary  Examiner  in  charge  of  the  concrete  division  of  the  Patent 
Office  in  numerous  motions  for  dissolution  in  interference  proceed- 
ings. Differentiation  between  the  case  under  consideration  and 
the  prior  art  cited  in  the  motion  to  dissolve  the  interference  is  effected 
by  considering  the  mode  of  operation  of  the  structure  as  a  machine 
or  mechanism  on  the  general  principles  elucidated  in  the  preceding 
chapters. 

The  word  " machine"  as  used  in  the  patent  statute  is  not  con- 
fined to  the  popular  idea  of  a  mechanism  consisting  of  pulleys, 
shafts,  levers,  etc.,  in  which  the  motion  is  quite  obvious,  but  is  to 
be  interpreted  in  accordance  with  the  broader,  general  definition 
of  machine,  as  given  in  Webster's  Dictionary,  as  follows- 

"Any  device  consisting  of  two  or  more  resistant,  relatively 
constrained  parts,  which,  by  a  certain  predetermined  intermotion, 
may  serve  to  transmit  and  modify  force  and  motion  so  as  to  produce 
some  given  effect  or  to  do  some  desired  kind  of  work.  According 
to  the  strict  definition,  a  crowbar  abutting  against  a  fulcrum,  a 
pair  of  pliers  in  use,  or  a  simple  pulley  block  with  its  fall,  would 
be  a  machine." 

In  applying  this  definition  it  is  evident  that  there  are  resistant 
parts  in  the  composite  structure  of  a  beam  or  slab,  to  wit:  the  steel 
and  the  concrete.  When  any  load  is  brought  upon  this  combina- 
tion, deformations  occur  in  both  steel  and  the  concrete  and  the  rela- 
tive motion  of  these  parts  is  constrained  by  the  shrinkage  grip  of 
the  concrete  on  the  steel  operating  thru  bond  shear.  This  com- 
bination performs  the  desired  work  of  carrying  the  load  to  the 


SCOPE    OF    PATENTS  421 

support.  It  modifies  the  motion  of  the  load  and  transmits  motion 
in  its  parts  until  equilibrium  is  established  between  the  load  and  the 
supporting  combination,  the  steel  and  the  concrete. 

That  the  movements  in  a  slab  or  beam  are  not  noticeable  to  the 
ordinary  observer  in  no  wise  affects  the  application  of  this  definition. 
We  might  state  for  example,  that  in  the  machine  or  mechanism  for 
transmission  of  sound,  known  as  the  telephone,  the  motion  of  the 
disc  is  too  slight  to  be  observed  by  the  eye.  We,  however,  readily 
measure  the  rapidity  and  extent  of  its  motion  by  the  sound  it  pro- 
duces. So  in  the  beam  or  slab  where  these  motions  are  enormously 
greater  than  they  are  in  the  telephone  disc,  they  are  readily  measured 
by  the  deflectometer  and  strain  gage. 

There  are  many  patents  upon  reinforcement  as  a  manufacture, 
per  se,  such  that  when  combined  with  concrete  in  the  finished  struc- 
ture they  do  not  produce  a  load-carrying  mechanism  differing  in 
any  wise  in  principle  from  others  which  have  preceded  them.  Such 
patents,  protect  an  invention,  the  utility  of  which  is  limited  to  more 
convenient  handling  of  the  materials  in  the  erection  of  the  structure 
or  more  economical  method  of  placing  the  material  in  the  desired 
position  in  the  finished  structure  without  the  introduction  in  any  wise 
of  anything  new  or  novel  in  the  mode  of  operation  of  the  structure 
itself. 

Scope  of  Patents.  The  " scope"  of  a  patent  or  its  power  to  secure 
to  its  owner  the  limited  monopoly,  or  control  of  the  invention  which 
forms  its  subject  depends  on  the  relation  of  the  invention  to  the  state 
of  the  art,  or  what  had  been  done  at  the  time  of  the  appearance 
of  the  invention,  and  also  upon  the  skill  with  which  the  claims  have 
been  drawn.  Having  reference  to  the  chronology,  or  the  time  re- 
lation of  the  invention  to  the  state  of  the  art,  a  patent  may  be 
basic,  or  " pioneer"  as  it  is  usually  termed,  or  it  may  be  a  specific 
or  narrow  patent.  Putting  it  otherwise,  a  patent  may  be  generic, 
that  is  for  a  genus;  or  specific,  that  is  for  a  particular  species  of  the 
genus  which  forms  the  subject  of  the  generic  or  pioneer  patent,  and, 
obviously,  this  relation  of  genus  and  species  requires  that  the  thing 
constituting  the  species  must  be  within  the  control  or  dominion  of 
the  genus  patent. 

Even  tho  a  given  invention  may  not,  in  the  most  general  aspect, 
be  new,  and  hence  a  pioneer  in  the  broadest  sense,  yet  by  virtue  of 
its  practical  value  and  importance  in  the  art  it  may  be  regarded  by 
the  courts  as,  in  a  sense,  a  pioneer  and  to  such  a  patent  the  courts 
have  applied  the  term  of  " limited  pioneer,"  and  the  scope  of  its 


422  INVESTIGATION    OF    THE    SCOPE    OF    PATENTS 

protection  is  broad.  Such  a  patent  may  be  regarded,  so  to  speak, 
as  a  sub-genus,  and  obviously  the  invention  thereof  can  exist  in 
the  form  of  various  species  falling  within  this  sub-genus  and  hence 
proper  to  be  dominated  by  the  limited  pioneer  patent. 

To  illustrate  the  matter  by  the  subject  in  hand,  at  the  time  of 
advent  of  the  "Mushroom"  or  true  flat  slab  type  of  reinforced  con- 
crete construction,  reinforced  concrete  was  an  old  thing  in  building 
construction.  The  "Mushroom"  invention,  therefore,  could  not 
be  covered  by  a  patent  which  would  dominate  or  control  any  and 
all  combinations  of  concrete  and  steel  combined  to  utilize  the  com- 
pressive  strength  of  concrete  and  the  tensile  strength  of  steel.  The 
patent  covering  it  could  not  be  a  pioneer  patent  in  the  broadest 
sense.  But  the  "Mushroom"  invention  being  the  first  instance  in 
the  art  of  a  true  continuous  flat  slab  resting  on  columns,  and  resist- 
ing flexure  by  circumferential  cantilever  action  about  the  head  of 
the  column  as  explained  in  Chapters  IV  and  V,  and  which  by  flexure 
between  the  columns  about  the  diagonal  center,  secures  plate  action 
by  wide  spreading  reinforcement  of  substantially  equal  strength  in 
all  directions,  a  patent  therefor  is  to  that  extent  a  pioneer,  and  is 
for  a  sub-genus  in  reinforced  concrete  construction,  and  all  sub- 
sequent patents  having  characteristics  of  the  sub-genus  are  merely 
species  thereof  and  hence  proper  to  be  dominated  or  controlled  by 
the  sub-genus  patent  if  the  claims  of  this  patent  are  finally  sus- 
tained by  decisions  of  the  Courts  as  they  have  been  in  repeated 
contests  in  the  Patent  Office. 

The  Importance  of  Investigating  the  Scope  of  a  Patent.  The  scope 
of  a  patent  is  determined  by  the  breadth  of  its  claims.  Any  limita- 
tions or  conditions  placed  in  its  claims  narrow  and  restrict  its  scope. 

The  scope,  and  consequently  the  value  of  a  patent  as  a  means 
of  controlling  a  given  construction,  while  depending  principally  on 
the  character  of  the  patent  claims,  is  also  affected  by  the  state  of  the 
art  at  the  time  of  the  advent  of  the  invention,  and  by  the  effect  of 
proceedings  in  the  Patent  Office  to  secure  the  claims.  A  patent 
attorney,  thru  a  misunderstanding  of  the  invention,  or  from  want  of 
skill  or  experience,  may  draw  or  word  the  claims  so  that,  tho  in 
fact,  the  invention  is  a  pioneer,  the  patent  itself  is  narrow  or  specific, 
and  limited  to  a  particular  species  of  a  genus,  instead  of  dominat- 
ing all  species  of  that  genus.  The  state  of  the  art  may  be  such  that 
the  field  for  a  new  construction  is  so  circumscribed  that  the  scope 
of  the  claims  may  be  restricted  even  to  the  identical  construction 
shown  in  the  drawings  of  the  patent.  At  this  point,  it  seems  proper 


PRIOR    ART  423 

to  explain  that  the  state  of  the  art  embraces  everything  relating  to 
the  subject,  whether  found  in  books  or  other  publications,  and  pat- 
ents here  and  abroad,  and  what  has  been  done  in  this  country  in 
actual  use. 

The  Patent  Office,  by  reason  of  its  limited  force  and  facilities 
rarely  does  more  than  search  thru  United  States  and  foreign  patents 
before  deciding  whether  or  not  to  grant  a  patent.  It  has  absolutely 
no  facilities  for  ascertaining  what  has  been  done  in  the  way  of  actual 
use,  and,  hence  innocently  and  excusably,  at  times  grants  patents 
for  what  has  already  long  been  in  actual  use,  but  such  patents,  of 
course,  have  no  validity. 

Revised  Statute,  Section  4886,  covers  this  point  in  the  following 
words : 

"A  patent  may  be  obtained  by  any  person  who  has  invented 
or  discovered  any  new  and  useful  art,  machine,  manufacture,  or 
composition  of  matter,  or  any  new  and  useful  improvement  thereof, 
not  known  or  used  by  others  in  this  country  before  his  invention  or 
discovery  thereof,  and  not  patented  or  described  in  any  printed  publi- 
cation in  this  or  any  foreign  country  before  his  invention  or  dis- 
covery thereof,  or  more  than  two  years  prior  to  his  application,  and 
not  patented  in  a  country  foreign  to  the  United  States  on  an  applica- 
tion filed  more  than  twelve  months  before  his  application,  and  not  in 
public  use  or  on  sale  in  the  United  States  for  more  than  two  years 
prior  to  his  application,  unless  the  same  is  proved  to  have  been 
abandoned,  upon  payment  of  the  fees  required  by  law  and  other 
due  proceedings  had." 

The  effect  of  proceedings  in  the  Patent  Office  and  the  importance 
of  looking  into  the  matter  when  determining  the  value  of  a  patent 
is  well  shown  in  a  recent  case.  A  patent  was  submitted  to  an  en- 
gineer for  purchase,  for  which  he  was  willing  to  pay  $50,000,  provid- 
ing it  was  what  it  was  purported  to  be.  An  examination  of  the 
application  and  the  records  of  the  Office  soon  disclosed,  that  by  reason 
of  amendments  and  disclaimers  filed  by  the  inventor  in  response 
to  rejections  of  the  Patent  Office,  the  claims  of  the  patent  were  of 
such  narrow  scope  as  not  to  cover  even  the  actual  commercial  form 
of  the  invention,  and  so  the  patent  was  valueless. 

Prior  Art.  The  art  necessarily  includes  all  prior  patents,  appli- 
cations, domestic  and  foreign,  and  all  domestic  use  prior  to  actual 
date  of  invention.  It  may  be  said  that  the  only  difference  between 
the  limitations  by  prior  art  and  anticipation  is  that  the  former  limits 
the  scope  of  the  claims  while  the  latter  kills  them  and  it  is  not  in- 
frequently a  fact  that  the  limitation  of  a  claim  by  the  prior  art  is 
such  as  to  all  intents  and  purposes  destroys  its  utility. 


424  RELATION    OF    GENUS    ABD    SPECIES    PATENTS 

Regarding  such  limitations,  the  Court  of  Appeals  of  the  Second 
Circuit  says: 

"Where  the  patentee  specifies  a  special  form  by  which  the 
effect  of  the  invention  is  produced  or  otherwise  confines  himself 
to  the  particular  form  of  what  is  prescribed  he  is  limited  thereby 
in  his  claims  for  infringement." 

And  in  the  Keystone  Bridge  Co.  case,  95  Fed.  U.  S.  274,  at  page 
278,  the  Supreme  Court  said: 

"They  (the  patentees)  cannot  expect  the  courts  to  wade  thru 
the  history  of  the  art,  and  spell  out  what  they  might  have  claimed 
and  have  not  claimed.  .  .  .  But  the  courts  have  no 
right  to  enlarge  a  patent  beyond  the  scope  of  its  claims  as  allowed 
by  the  Patent  Office.  .  .  As  patents  are  procured  ex  parte, 
the  public  is  not  bound  by  them,  but  the  patentees  are.  Ard 
the  latter  cannot  show  that  their  invention  is  broader  than  the 
terms  of  their  claim,  or,  if  broader,  they  must  be  held  to  have 
surrendered  the  surplus  to  the  public." 

Genus  and  Species  Patents.  *  It  seems  desirable  to  correct  a  wide- 
spread error  as  to  the  scope  of  the  grant  of  the  patent  by  the  Patent 
Office  in  its  relation  to  other  patents  (either  earlier  or  of  later  date 
of  issue)  as  far  as  the  right  to  use  the  construction  of  such  patent  is 
concerned. 

The  grant  of  a  patent  confers  no  right  to  use  the  construction 
shown  in  the  patent.  It  simply  gives  the  right  to  the  owner  of  the 
patent  to  prevent  others  from  using  that  construction.  The  legal 
proposition  may  be  illustrated  in  this  way : 

A  patent  is  issued  in  1910  to  C  on  a  given  type  of  construction. 
All  that  this  patent  gives  to  C  is  the  right  to  stop  other  persons 
from  making  use  of  or  selling  the  construction  set  forth  in  the  claims 
of  that  patent.  It  gives  no  such  right,  as  the  right  to  use 
the  subject  matter  of  the  patent  and  the  patent  office  has  no  authority 
in  law  to  give  the  right  actually  to  use  the  construction  shown  in 
the  patent  for  the  reason  that  because  of  some  dominating  genus 
patent  the  owner  of  the  latter  B  has  the  right  to  prevent  others 
from  using  the  construction  which  forms  the  subject  of  the  patent 
to  C  to  which  patent  that  of  C  stands  in  the  relation  to  that  of  B 
of  a  species  to  a  genus.  The  fact  that  the  construction  of  C  and  B 
belong  to  the  same  family,  the  B  genus,  in  reality  are  merely  differ- 
ent species  thereof  or  different  expressions  of  the  same  idea,  explains 
the  relation  of  a  broad  to  a  specific  or  narrow  grant. 

Consequences  of  Infringement.  For  the  infringement  of  a  patent, 
the  law  provides  redress  in  two  forms:  One  in  compensation  in 
money,  which  covers  the  profits  made  by  the  infringer  and  the 
*Reinforced  Concrete  Patents,  Williamson. 


INFRINGEMENT  425 

damages  (which  the  Court  may  treble)  to  the  owner  of  the  patent, 
together  with  the  costs  of  the  suit.  The  other  is  an  injunction 
prohibiting  further  infringement,  which  in  the  case  of  a  building 
would  be  the  prohibition  of  further  use  of  it.  Among  the  persons 
liable  for  infringement  in  the  case  of  a  building  are  the  builder,  the 
owners  and  the  user. 

Indeed,  the  Courts  have  gone  so  far  as  to  order  the  destruction 
of  the  infringing  thing.  Thus  the  Supreme  Court  of  the  United 
States  in  Birdsell  v.  Shaliol,  112  U.  S.  485,  said: 

"But  an  infringer  does  not,  by  paying  damages  for  making 
and  using  a  machine  in  infringement  of  a  patent,  acquire  any 
right  himself  to  the  future  use  of  the  machine.  On  the  contrary, 
he  may,  in  addition  to  the  payment  of  damages  for  its  infringe- 
ment, be  restrained  by  injunction  from  its  further  use,  and  when 
the  whole  machine  is  an  infringement  of  the  patent,  be  ordered  to 
deliver  it  up  to  be  destroyed." 

Some  idea  of  the  great  favor  which  the  law  gives  to  the  owner 
of  a  patent  in  enforcing  his  rights  may  be  gathered  from  a  few 
decisions  of  the  courts.  The  Court  of  Appeals  of  the  Seventh 
Circuit  speaks  of  the  patent  owner  as  a  "czar,"  so  great  is  his  power 
under  the  law,  and  other  Courts  describe  his  power  in  language 
equally  as  strong.  Said  that  Court  of  Appeals  (which  sits  at  Chicago) 
in  Victor  Talking  Machine  Co.  v.  The  Fair,  123  Fed.  Rep.  424: 

"'All  that  the  government  can  and  does  grant,  is  the  right  to 
exclude  others  from  practicing  his  invention  without  his  consent. 
Within  his  domain,  the  patentee  is  czar.  The  people  must  take 
the  invention  on  the  terms  he  dictates,  or  let  it  alone  for  seventeen 
years.  This  is  a  necessity  from  the  nature  of  the  grant. 

The  field  being  his  own  property  and  there  is  no  law  for 
seizing  it  and  adjudging  his  damages,  he  cannot  be  compelled  to 
part  with  his  own  except  on  inducements  to  his  liking." 

Said  the  Court  in  General  Electric  v.  Wise,  119  Fed.  Rep.  922: 

"No  time  will  be  used  in  answering  this  suggestion,  except  to 
say  that  if  complainant's  patents  are  valid,  it  is  entitled  to  protec- 
tion by  injunction  against  all  the  world.  No  other  person  or 
company  can  use  its  property  of  this  description  without  its  consent 
and  relegate  it  to  an  action  for  damages.  If  this  patent  is  valid 
complainant  has  an  absolute  right  under  the  laws  of  our  country 
to  the  use  of  the  patent  and  to  designate  the  parties  on  whom  it 
will  confer  the  right  to  use  it." 

While  the  preceding  statements  clearly  illustrate  the  position 
taken  by  the  Court  in  the  case  of  machines  used  to  turn  out  a  product, 
the  decisions  are  less  numerous  and  are  somewhat  conflicting  with 
reference  to  structures  or  load-carrying  machines.  In  a  building, 
looking  at  it  as  a  load-carrying  mechanism  from  the  viewpoint  taken 
by  the  Supreme  Court  in  the  Birdsell  v.  Shaliol  case,  the  Court  took 
the  stand  that  the  infringer  may  be  restrained  by  injunction  from 


426  RECOVERY    FOR    INFRINGEMENT 

the  further  use  of  it  as  a  machine  and  when  the  whole  machine  is  an 
infringement  of  the  patent,  to  order  him  to  deliver  it  up  to  be  des- 
troyed. Here  the  Court  seems  to  apply  what  was  termed  the  "Rule 
of  Reason"  which  created  much  discussion  in  the  rulings  relative 
to  the  scope  of  the  Sherman  law. 

While  no  ultimate  conclusions  have  been  reached  in  the  cases 
involving  building  construction  where  patents  have  been  sustained, 
this  rule  would  indicate  they  should  be  so  construed  as  to  do  justice 
to  the  holders  of  such  patents  and  to  fairly  carry  out  the  contract 
entered  into  by  the  government  when  it  issued  such  patents. 

In  view  of  the  fact  that  the  finished  building  contains  much 
more  than  the  mere  load  carrying  skeleton  or  mechanism  of  reinforced 
concrete,  an  injunction  against  its  use  without  qualification  would 
be  inequitable  to  the  owner  and  in  a  measure  unreasonable.  A 
choice  in  extreme  cases  between  an  injunction  and  the  payment  of 
three  times  a  contractor's  profit  of  fifteen  percent  of  the  value  of  the 
cement  work  in  the  case  of  contested  cases  and  of  the  usual  fee  in 
cases  not  contested  would  seem  to  be  the  limit  of  reasonable  protec- 
tion to  which  the  holder  of  a  broad  patent  may  be  entitled,  even  tho 
this  amount  might  not  pay  in  a  single  instance  for  the  expense  of 
carrying  a  suit  to  conclusion. 

In  cases  where  the  patent  covers  merely  a  convenient  form  of 
make-ready  for  the  reinforcement  without  involving  a  new  mode 
of  operation  differing  materially  from  other  and  older  forms,  it  would 
seem  that  the  redress  should  be  logically  limited  to  a  judgment  for 
damage  to  be  recovered  from  those  making  or  putting  up  the  struc- 
ture who  have  been  directly  benefited  by  the  economy  resulting  from 
the  form  of  manufactured  material  used.  Certainly  in  this  second 
class  of  inventions,  classification  as  a  new  and  novel  load-carrying 
device  or  mechansim  would  not  hold  good. 

Were  an  injunction  unqualified  as  suggested  issued  against  the 
owner  of  an  infringing  structure,  unconditionally,  it  would  place 
the  owner  in  the  serious  position  of  being  forced  to  pay  any  amount 
which  the  patentee  might  demand.  The  inequity  of  such  a  position 
as  this  has  apparently  deterred  some  members  of  the  Judiciary 
from  deciding  in  favor  of  the  patent.  One  Federal  Judge  made  the 
assertion  that  he  would  not  issue  an  injunction  which  apparently 
was  the  only  remedy  because  in  the  case  of  a  building  he  did  not 
consider  it  to  be  equitable  and  construed  the  claims  of  the  patent  in 
an  extremely  narrow  manner,  which  decision  enabled  him  to 
escape  the  dilemma. 


AGGREGATION    VS.    COMBINATION  427 

The  Court  of  Appeals  was  apparently  dissatisfied  with  this 
decision  when  it  was  confronted  with  similar  considerations.  In- 
stead of  upholding  the  decision  of  the  Lower  Court  on  the  grounds 
on  which  it  was  rendered  it  stated  that  in  its  opinion  a  concrete 
floor  slab  was  merely  an  aggregation.  In  other  words,  the  bending 
resistance  of  the  slab  was  the  sum  of  the  bending  resistance  of  the 
steel  and  the  concrete  acting  separately.  In  this  decision,  altho 
furnished  with  a  complete  library  of  all  American  treatises  on  rein- 
forced concrete  the  Court  over-looked  the  connecting  link  between 
the  concrete  and  the  metal  known  as  adhesion  or  bond  which  causes 
the  two  materials  to  act  together  and  form  a  true  combination. 
This  legal  point  may  need  some  discussion. 

Referring  to  Curtiss  Fixed  Law  of  Patents: 

"AGGREGATION:  The  distinction  between  an  aggregation 
and  a  true  combination  is  not  always  clear.  The  main  test  lies 
in  examination  of  the  result — the  function  performed.  If  that 
result  is  the  sum  of  the  several  actions  of  the  elements,  it  is  an 
aggregation;  if  it  is  the  product  of  those  actions — if  the  action 
of  one  element  so  modifies  the  action  of  another  that  the  resultant 
action  differs  from  the  sum  of  the  separate  actions — it  is  a  true 
combination." 

The  Circuit  Court  of  the  Eighth  Circuit,  No.  3801,  thus  explains 
the  difference  between  an  aggregation  and  a  combination. 

"For  example,  it  is  not  invention  to  take  a  fire  pot  from  an 
old  stove,  a  flue  from  another  and  a  coal  reservoir  from  a  third 
and  assemble  them  where  each  merely  performs  its  old  function 
in  its  new  location."  Hailes  v  Van  Wormer,  20  Wall.  353. 

The  error  of  the  view  that  this  is  the  state  of  the  case  with  a  concrete 
beam  or  slab  may  be  illustrated  as  follows:  Consider  the  case  of 
two  planks,  one  superimposed  upon  the  other,  and  load  them  in 
this  position;  then  the  resistance  supplied  by  the  two  planks  in 
bending  under  load  is  the  aggregate  resistance  of  the  two  planks 
and  is  accompanied  by  the  phenomena  of  the  lower  corners  of  the 
upper  plank  sliding  by  the  upper  corners  of  the  lower  plank,  as  the 
planks  bend  under  load.  In  this  case  the  bending  resistance  of  the 
planks  is  the  aggregate  of  the  bending  resistance  of  the  two  elements, 
and  this  phenomena  of  sliding  must  occur  when  the  connecting  link 
of  bond  shear  is  lacking  between  the  two  planks.  But  when  by 
bolting  and  glueing  the  two  planks  together  so  that  sliding  is  pre- 
vented we  secure  shearing  resistance  between  them,  the  stiffness  of 
the  two  planks  so  joined  becomes  four  fold  the  aggregate  stiffness 
of  the  two  under  load  and  the  strength  is  increased  one  hundred 
percent.  With  this  arrangement  the  two  planks  no  longer  form  an 
aggregation,  they  have  become  a  combination  with  greatly  in- 


428  PATENT    OFFICE    PRACTICE    VS.    SOME    COURT   DECISIONS 

creased  efficiency  as  a  load  carrying  mechansim,  and  were  the  com- 
bination novel  to  the  art  would  be  patentable. 

Now  we  have  pointed  out  in  earlier  chapters  that  the  joint  action 
of  the  two  materials  in  a  Mushroom  slab  thru  bond  shear  produces 
a  result  widely  different  from  the  aggregate  resistance  of  the  two 
elements  and  the  same  quantities  of  the  two  elements  have  widely 
different  efficiencies  dependent  upon  the  manner  and  arrangement 
of  the  reinforcement  in  the  matrix  vertically  and  horizontally, 
which  determines  the  law  or  mode  of  operation  of  the  structures. 
This  consequently  is  a  combination  and  patentable,  as  held  by  the 
experts  of  the  Patent  Office. 

In  the  decision  handed  down  by  the  Lower  Court  in  an  Eastern 
Circuit,  the  Judge  concluded  that  a  patent  on  a  reinforced  concrete 
structure  could  only  be  granted  as  a  patent  for  an  art.  The  art 
of  burying  metal  in  concrete  being  older  than  Christian  civilization, 
if  this  decision  is  concurred  in  by  the  higher  courts,  the  United 
States  Patent  Office  is  placed  in  the  position  of  having  accepted 
fees  in  a  wholesale  manner  for  patents  on  a  branch  of  science  or 
industry  not  patentable  under  the  Constitution,  and  the  trained 
experts  of  the  Patent  Office  are  open  to  criticism  not  for  having 
erred  excusably  in  granting  patents  for  what  had  already  been  in 
actual  use,  but  for  inaugurating  a  policy  of  accepting  fees  upon  an 
industry  which  the  ruling  of  the  Courts  holds  to  be  an  improper 
subject  matter  for  patent. 

Looking  at  the  decision  of  the  Court  of  Appeals,  referred  to  in 
another  light,  if  the  strength  of  concrete  and  reinforcement  is  an 
aggregation  and  bond  shear  plays  no  part  in  the  mode  of  operation 
of  the  structure  then  the  technical  men  of  the  Patent  Office  are  in 
error  in  granting  patents  on  alleged  improvements  which  must  from 
their  very  nature  as  an  aggregation  be  absolutely  useless. 

This  conclusion  would  seem,  however,  to  be  incorrect,  for  the 
reason  that  the  Examiners  in  Chief  and  the  Commissioner  of  Patents 
inaugurating  this  practice,  are  trained  experts  in  their  particular 
branches,  while  the  training  of  the  Federal  Judiciary  is  along  legal 
rather  than  along  technical  lines.  Further,  the  hundreds  of  millions 
of  dollars  of  reinforced  concrete  construction  show  that  it  is  com- 
mercially valuable  and  useful.  Any  opinion  which  involves  a 
hypothesis  to  the  contrary  is  accordingly  erroneous  and  untenable. 

In  the  specific  instances  cited,  while  these  decisions  are  not  in 
accord  with  the  general  tenor  of  rulings  by  Federal  Courts,  never- 


DIFFICULT    DUTIES    OF    FEDERAL   JUDGES  429 

theless,  a  comparison  of  the  efficiency  of  the  present  method  of 
court  procedure  in  deciding  technical  causes  where  the  judge  is 
trained  along  legal  lines  in  contradistinction  to  one  trained  along 
technical  lines  in  the  particular  branch  under  which  the  case  comes 
may  not  be  amiss. 

One  engineer  thus  describes  the  operation  of  the  Federal  Courts: 
After  two  years'  time,  at  an  expense  of  fifteen  to  twenty  thousand 
dollars,  the  probability  is  that  the  litigant  will  secure  in  a  com- 
plicated case,  a  decision  from  some  Court  of  Appeals  which  by 
implication  conveys  to  the  expert  the  unavoidable  conclusion  that 
the  law  of  gravitation  and  conservation  of  energy  is  held  by  the 
Court  to  be  inoperative  or  unconstitutional  in  this  particular  branch 
of  science.  Perse verence  and  continued  effort  for  two  or  three  years 
more  may  secure  an  opposing  decision  in  another  circuit  and  then 
by  carrying  the  matter  to  the  Supreme  Court  of  the  United  States, 
in  the  course  of  eight  or  ten  years  from  the  date  of  filing  the  original 
suit,  a  decision  will  very  likely  be  rendered  in  accordance  with  fixed 
natural  laws. 

The  judiciary,  however,  are  not  really  to  be  blamed  for  this 
state  of  affiairs.  They  perform  their  appointed  task  as  best  they 
may. 

Should  a  structural  engineer  be  requested  by  his  employer, 
first  to  report  on  a  complicated  question  of  inorganic  chemistry, 
then  upon  a  metallurgical  question,  and  then  on  a  mechanical 
proposition  entirely  outside  of  his  special  line  of  business,  in  the  same 
month,  he  would  decline  to  undertake  the  work  and  conclude  that 
his  employer  was  non  compos  mentis,  and  resign  his  engagement. 
The  employer  of  the  Federal  Judges  in  this  case,  is  the  United  States 
Congress  which  has  exhibited  in  its  provision  for  the  adjudication 
of  technical  cases  a  lack  of  capacity  in  keeping  with  its  lack  of 
technical  knowledge.  The  system  inaugurated  by  our  Congress 
is  incomparably  bad.  It  provides  the  same  judge  or  man  learned 
in  law  to  try  a  case  involving  complicated  questions  of  structural  de- 
sign as  to  decide  complicated  questions  in  the  chemistry  of  dyes.  It 
provides  a  judge  qualified  by  learning  in  law  only  to  decide  metal- 
lurgical questions  and  to  decide  all  manner  of  special  technical  and 
mechanical  questions  which  it  would  seem  must  try  his  patience  to  the 
extreme,  for  he  is  supposed  to  find  the  time  to  study  and  digest  all 
the  scientific  principles  that  may  be  involved  in  the  question  at 
issue  presented  by  the  one  side  and  the  clever  misleading  evidence 


430  TECHNICAL    AND    NON-TECHNICAL    JUDGES    COMPARED 

of  experts  on  the  other  side,  specially  designed  to  confuse  rather 
than  clear  up  the  question  of  fact  at  issue. 

In  this  connection  a  comparison  of  the  relative  efficiency  of  a 
scientific  expert  as  a  judge,  with  the  judge  qualified  by  ordinary 
legal  training  will  emphasize  the  point  above  made. 

The  discussion  of  the  entire  prior  art  and  the  technical  prin- 
ciples involved  are  brought  into  question  in  the  course  of  an  inter- 
ference by  a  motion  for  dissolution  in  the  Patent  Office.  Such  a 
motion  is  usually  disposed  of  in  from  three  to  six  hours  argument 
before  the  expert  examiner  as  judge.  It  is  needless  to  say  that  no 
principles  of  elementary  mechanics  nor  any  digest  of  the  theo- 
retical principles  need  be  presented  to  such  a  judge.  He  would  be 
thoroly  familiar  with  those  principles,  and  hence  in  two  to  four 
hours  the  same  ground  would  be  covered  as  would  be  covered  in  two 
to  four  years'  time  in  deciding  the  same  questions  before  the  Federal 
Judiciary. 

The  relative  efficiency  from  the  standpoint  of  cost  may  be  com- 
pared without  exaggeration  by  saying  that  where  it  would  cost 
dollars  in  deciding  a  question  before  a  technical  judge,  it  costs  an 
equal  number  of  thousands  of  dollars  to  decide  the  points  at  issue 
before  the  Federal  Judiciary  in  any  case  which  involves  mechanical 
principles  that  are  in  the  least  complicated. 

Thus  by  establishing  an  unscientific  system,  Congress  has  to 
a  large  degree  nullified  the  intent  of  the  Constitution  in  its  attempt 
to  encourage  the  poor  but  worthy  inventor  and  by  its  provisions 
for  his  benefit,  has  rendered  a  patent  a  luxury  for  the  wealthy  only, 
a  privilege  entitled  to  respect  only  in  proportion  to  the  bank  account 
of  the  holder.  The  wealthy  piratical  infringer  under  the  present 
system  expects  to  wear  out  the  deserving  inventor  by  the  expense 
which  is  necessarily  involved  in  carrying  out  this  cumbersome 
system  of  enforcing  his  rights,  and  in  the  opportunity  which  this 
system  offers  for  bringing  fake  suits  on  patents  which  are  worth- 
less and  merely  alleged  to  apply  to  the  subject  matter  of  the  im- 
provement for  the  purpose  of  crippling  the  inventor  from  the  financial 
standpoint. 

As  tho  the  apparent  total  absence  of  systematic  provision  for 
rendering  just  judicial  decisions  of  technical  questions  as  outlined 
in  the  preceding  discussion  were  in  fact  lacking  in  some  particulars 
that  might  prevent  it  from  being  as  bad  as  could  possibly  be  devised, 


CUMBERSOME    METHODS    OF    DECIDING    PATENT    SUITS  431 

Congress  has  provided  that  there  shall  be  nine  different  independent 
federal  circuit  courts  each  to  be  local  and  provincial  in  its  char- 
acter, and  omnipotent  in  its  own  district.  But  the  decisions  of  no 
one  court  are  of  binding  force  in  any  other  district.  No  patent 
court  of  appeals  exists  which  may  render  a  decision  in  a  single  suit 
once  and  for  all  for  the  country  at  large  in  the  interest  of  simpli- 
fication of  legal  controversies  about  patents,  except  in  case  of  con- 
flicting decisions  as  previously  stated.  This  arrangement  tho  well 
devised  for  the  emolument  of  the  legal  fraternity  is  objectionable 
in  the  extreme  from  the  standpoint  of  the  interests  of  the  common 
citizen. 


432 


INDEX 


ADHESION  (See  Bond)  Page 

AGGREGATE 

Broken  Stone 13,  15 

Coarse  aggregate 15 

Gravel 13 

Proportions 14,  347 

Sand 11-13 

Specifications  for 11 

ANALYSIS 

Cement 8-11,  16 

Sand 11-13 

Steel 34-38,  40 

Strength  of  Concrete 14-17,  21,  68,  69 

ARCH  ACTION  IN  SLABS  (See  Slabs) 

BARS 

Deformed  bars 18-21 

Effect  of  shape  of  bar  on  adhesion 18 

Quality,  how  determined 36-37 

Rust  of  bars 18 

Specification  for 34-36 

BEAMS 

Classification 95 

Continuous 96-101 

Cottancin  Beam  Construction 90,  119-120 

Diagram  of  k  and  j  curves 79 

Diagonal  tension 87-93,  112 

Economic  design 101-103 

Formulas  for 74-78 

Hennebique  type 98 

Limiting  steel  ratios  of  deep  and  shallow  beams 81-87 

Percentage  of  reinforcement 80-96 

Position  of  neutral  axis 81-82 

Reinforced  for  compression 77,96 

Standard  notation 73,  74 

Shear 87,  108-112 

Stirrups 117 

Turner  type 97 

BEARING  VALUE  OF  SOIL  (See  Foundations) 

BEARING  WALLS 346 

BENDING  MACHINES 38-40 

BENDING  MOMENT 

As  affected  by  continuity  in  beams  and  in  slabs 148-150 

Experimental  results  illustrating  same 148-155 

Fundamental  relations  of  moment  magnitudes 146 

In  four-way  slabs  at  mid  span  and  supports 146,  147 

In  beams 97-101 

Relation  of  bending  moments  and  shears 108,  287 

True  and  apparent  moment 166,  167 


INDEX  433 

BOND  Page 

Between  concrete  and  steel 17-21 

Between  concrete  and  different  shaped  bars 17,  18 

Between  rough  slab  and  finish  coat  or  strip  fill 389 

Deformed  bars 20,  21,  132-133 

BOND  SHEAR 

In  blocks 121-123 

In  splices 123-124 

In  beams 124-127 

In  slabs 127-130 

Mechanics  of 130-132 

CALCULATIONS 

Calculations  of  stresses  and  deflections  verified  by  tests 

Bridge  over  Tischers  Creek,  Duluth 223-224 

Comparison  of  Norcross  and  Mushroom  slabs 253-265 

Curtis  Street  Bridge,  Denver 225,  226 

Deere  and  Webber  Co.  Building.,  Minneapolis 217,  229 

Ford  Building,  Los  Angeles 221,  223 

Larkin  Building,  Chicago 219,  230,  231 

Mushroom  slab 246-253 

Norcross  slab 236-246 

Northwestern  Glass  Co.  Building 227,  228 

Rectangular  panels  supported  on  beams 285-295 

St.  Paul  Bread  Company  Building 185,  232-235 

Slabs  resting  on  beams 293-295 

CAST  STONE 380,  407,  408 

CEILING,  SUSPENDED 396 

CEMENT 

Composition 9 

Influence  of  fineness 9 

Proportions  in  concrete 14 

Quick  tests 9 

Soundness 10 

Standard  specification 8,  9 

Storage  of 11 

CENTERING 

Column  forms 45,  46 

Design  of 43 

Kind,  wood  or  sheet  metal 45 

Partial  removal  of  forms 32-34,  47,  48 

Proportioning  posts  and  joists 42 

Saving  in  centering  by  using  rich  concrete 347 

Sheet  metal  forms 45-47 

Wood  forms 41 

COLUMNS 

Bach's  tests 311-316 

Concrete  and  structural  columns  compared 327 

Considere  formula 317-318 

Economic  design  of 329 

Effect  of  vertical  reinforcement  and  hooping 306-307,  320,  321 

Formula  based  on  kind  of  bearing 324 

Formula  for  WTithey's  Columns 324,  325 

General  types 303-306 

Mode  of  operation  of  vertical  steel 320 

Necessary  fireproofing 364 

Phoenixville  tests 307-310,  316 

Pouring  columns 25,411 

Safe  upper  limit  of  working  stress 319 


434  INDEX 

COLUMNS— Continued  Pago 

Spacing 343 

Structural  columns  with  concrete 327 

Temperature  effects  on  columns 329 

Wall  columns 328,  329 

Withey's  tests 322,  323 

Working  stresses 326 

COMPARISON 

Beam  theory  and  slab  theory 237 

Concrete  and  structural  columns 327 

Concrete  and  terra  cotta 368 

Continuity  in  beams  and  slabs 148-151 

Cost  of  summer  and  winter  work 362 

Deflection  of  diagonal  and  direct  belts 145 

Fire  resistance  of  concrete  and  tile 368-371 

Limiting  steel  ratios,  deep  and  shallow  beams 84-88 

Limiting  steel  ratios,  thick  and  thin  slabs 295,  296 

Moment  curve  in  slab  and  beam 135 

Norcross  and  Mushroom  slab  tests 236 

Slab  action  and  arch  action 65,  264,  265 

Stiffness  of  beam  and  slab 139-144 

Test  data  for  columns 

Two-way  and  four-way  slabs 276,  277 

COMPUTATION 

By  proportion 53-55 

By  the  theory  of  work 115,  116 

By  equilibrium  of  infinitesimal  elements  of  slab 170-172 

CONCRETE 

Aggregate 11-15,  347 

Artistic  finish 

Stipple  coat 397 

Plaster  coat 398 

Brushing  and  washing 399-405 

Tooling 406 

Compressive  strength 14-17,  21,  68 

Coefficient  of  expansion 17,  114 

Consistency 24 

Crazing,  hair  cracks,  etc 379,  380 

Curing 15,  16,  409,  410 

Deformation  under  repeated  loading 70-72 

Elastic  properties 69-73,  83 

Effect  of  temperature  and  moisture 21,  22,  32 

Effect  of  oil,  grease,  etc 381,  382 

Electrolysis  of 383,  387 

Fireproof  properties  of ' 364-368 

Fire  resistance  of  concrete  and  tile  compared 368-371 

Fire  tests 386,  368,  370 

Floor  finish 388-391 

Hardening  at  different  temperatures , 25-34 

Hardening  compounds 390 

Improper  materials 13,  377,  378 

Method  of  handling  concrete  in  warm  and  cold  weather 27-34 

Mixed  dry  and  tamped 379 

Mixing 22,23 

Partial  removal  of  forms 32 

Permanence  of  concrete  structures 376 

Pouring,  proper  and  improper  methods 24,  25,  411 

Protection  of  steel  by  concrete  from  corrosion 375,  376 

Shearing  strength Ill,  112 

Splices  and  precautions  in  cold  weather 25,  31 

Temperature  cracks 381 


INDEX  435 

CONCRETE  (Continued)  Page 

Tensile  strength 69 

Test  for  hardness  before  removal  of  forms 25,  26,  33,  412 

Use  of  salt  in  concrete 28,  30 

Removal  of  forms  (proper  time  for) 30-34 

CONDUITS 

Proper  placing 396 

Provision  for  pipes,  plumbing,  etc 393 

CONSTRUCTION 

Execution  of  work 409 

Hardening  of  concrete 409-410 

Pouring  concrete 411 

Responsibility  of  the  engineer 412-415 

Responsibility  of  constructor  and  engineer  superintendent 416 

COST 

Adaptability  of  concrete 348 

Advantage  of  concrete  for  heavy  buildings 348 

Advantage  in  lower  cost  for  fire  resistance 348 

Advantages  of  slab  over  beam  construction 343,  344 

Analysis  of  items  of  cost,  materials,  labor,  etc 351-355 

Bearing  walls  vs  full  concrete  skeleton 346 

Bending  steel 357 

Centering,  relative  cost  of  different  types 345,  357-361 

Cinder  concrete 347,  377 

Concrete  vs  brick  exterior  walls 346 

Column  spacing 343 

Column  hooping 357 

Dead  charges 362 

Economic  column  design 330 

Economy  by  use  of  rich  mixture 347 

Economy  in  selecting  aggregate 347 

General  data 362,  363 

Rapid  erection 351 

Resistance  to  shock .  .  349,  350 

CRACKS 

Shrinkage 379,  380,  419 

Significance  of 379,  381,  416-418 

Temperature 381 

DEFLECTION 

Deflection  of  diagonal  and  direct  belts,  four  way  flat  slabs 145 

Deflection  of  side  beams 290 

Effect  on  deflection  of  finish  coat 66,  154 

Elastic  and  inelastic  deflection,  sag 22,  26,  33,  34,  56,  58 

Examples  of  computation  of — 

Hamm  Brewing  Co.  Stock  House,  St.  Paul 60 

Hoffman  Building,  Milwaukee 61 

John  Deere  Plow  Company  Bldg.,  Omaha 61 

Minneapolis  Knitting  Co.  Bldg 63 

Minneapolis  Paper  Company  Building 62 

Smythe  Block,  Wichita,  Kas 63 

State  Prison  Factory,  Stillwater,  Minn 60 

Formula  for  two  way  slab  supported  on  girders 61,  62,  292,  293 

Permissible  deflection  of  floor  slab. .  .  .  . 297,  298 

ELASTICITY 

Limits  for  steel 35,  37 

Limits  for  concrete 69,  72 

Limits  for  coulmns  at  yield  point .  .  324,  325 

Modulus  of  elasticity  of  concrete 70 

Modulus  of  elasticity  of  steel 68 


436  INDEX 

EXPANDED  METAL  (See  Reinforcement)  Page 

EXPANSION 

Coefficient  for  concrete 17 

Coefficient  for  steel 17 

Joints  when  needed 330 

EXPANSION  JOINTS 330 

EXECUTION  OF  WORK  (See  Construction) 

FACTOR  OF  SAFETY 

Nominal 103,  104 

True .  .  .103,104 

FINISH 

Artistic  surfaces 397-406 

Cement  floor  fiinish 389 

Hardening  Compounds 390 

Mixture  of  concrete  finish 390 

Paint  or  varnish 391 

Plaster  on  concrete 394,  395 

Wood  floor  (proper  widths  of  flooring) 66,  67,  388 

Strip  fill 66,  67,  388 

FIREPROOFING 

Thickness  of,  for  steel .364 

FIREPROOF  PROPERTIES  OF  (See  Concrete) 

FLOORS 

Beam  and  slab 61-65,  217,  285-291 

Calculation  of  deflection  (See  Calculation) 

Flat  slab  (See  slab) 

Finish  (See  finish) 

Permissible  deflection 297 

Relation  of  span  to  strength 54,  55 

Relation  of  thickness  to  strength 53,  54 

Strip  fill 388 

Test  loads 105-107 

Tile  and  concrete 299-300 

Types .  .  49-52 

FOOTINGS  (See  Foundations) 

FOUNDATIONS 

Bearing  on  soil,  stone,  etc 331-342 

Column  footings 332,  333 

Piles 

Advantages  of  concrete  piles  . 340 

Cast  in  place,  Raymond 335,  336 

Corrugated  piles 339 

Driven 333,  334 

Safe  bearing  (formula  for) 341,  342 

Simplex. .  334,  337,  338 

GRAVEL  (See  Aggregate) 

HARDENING  COMPOUNDS 390 

HEATING 

Heating  materials  for  concrete  when  necessary 28,  29,  31 

Provision  for  pipes 393,  394 

HISTORICAL  SKETCH 

Development  reinforced  concrete 1-6 


INDEX  437 

INSURANCE  Page 

Rates  on  building 371-374 

Rates  on  contents 373 

Rates  where  sprinkler  systems  are  used 372,  374 

Rates  compared  with  timber  construction 373 

INSULATION  OF  ROOFS 392 

JOINTS 

Expansion 25,  329 

In  beams 25,  31 

In  slabs 25,  31 

Proper  and  improper  methods  of  making  joints 25,  330 

LIMITS 

Deflection  permissible 297,  298 

Steel  ratios 80-96 

Ratio  of  depth  to  span  for  slabs 296 

Unit  stresses  in  columns 319 

LINES  OF  INFLECTION 152,  153 

MACHINES 

Bending 38-40 

Mixing 22,  23 

Overhead  Hoppers ' 23 

MOMENTS 

Determining  moment,  concrete  or  steel 78,  83,  295 

Fundamental  relation  of  moment  magnitudes 146 

Relation  of  maximum  moment  to  shear 134 

NOTATION 

For  flat  slabs  supported  on  beams 285,  292 

For  Mushroom  system 162-164 

For  proportion 55-56 

Standard  for  beams 73-74 

PATENTS 

Encouragement  to  progress  by 418,  431 

Genus  and  species  patents 424 

Importance  of  investigating  scope  of 421-423 

Consequence  of  infringement 425 

PLUMBING 

Provision  for 393 

Protection  of  lead  pipes 393 

PROPORTION 

Cement  and  aggregate  (See  Concrete) 

Strength  by  (See  Computation) 

PILES  (Concrete) 

Considere 334 

Gilbreth .' 338,  339 

Hennebique 

Raymond 334-336 

Simplex 334,  337,  338 

Safe  loads  for 341,  342 

REINFORCEMENT 

Expanded  metal .  .301,  302 

Lap  over  supports 412 

Size  and  spacing  of  rods 

Specification  for  steel 34-36 

Steel  ratios  in  beams 79-87 

Steel  ratios  in  slabs 296 

Tile  construction 229,  300 

Wire  fabric .  .301,  302 

SAND 

Analysis  of 11 

Influence  of  size  of  grains 11,  12 

Specification  for 11,  13 


438  INDEX 

SHEAR  Page 

Allowable  shearing  stresses  in  beams Ill 

Allowable  shearing  stresses  in  slabs 207 

Around  the  cap 206,  207 

Difference  in  continuous  and  simple  beams Ill 

Horizontal  in  beams 87 

Punching  shear  in  beams 108-110,  112 

Resistance  to  shear  of  reinforcement  around  cap 208,  209 

Resistance  of  reinforcement  to  shear  in  continuous  beams 98 

Resistance  of  reinforcement  to  shear  in  simple  beams 117 

Strength  in,  of  concrete 11,  112,  207 

SLABS 

Apparent  and  true  bending  moments 165-167 

Approximate  theory,  four-way  slab  supported  on  columns 269-274 

Approximate  theory,  two-way  slab  supported  on  columns 274-276 

Arch  action  in 65 

Computation,  method  of  (See  Computation) 

Effect  of  width  of  belts 135 

Example  of  calculation  of  stresses  and  deflection  verified  by  test 

(See  Calculation) 

Flat  slabs  supported  directly  on  columns,  general  Mushroom  type . 

152,  164,  213-215 

Flat  slabs  supported  on  columns,  modified  standard  type 

(a)  Concrete  Steel  Products  Design 215 

(b)  Taylor  and  Thompson  design 216 

(c)  With  depressed  heads 156,  157 

Flexure  and  strength,  theory  of 158,  161 

General  differential  equation  for  deflection 172-174 

General  differential  equation  for  moment 170-172 

General  differential  equation,  solution  of 174-177 

Notation  for  standard  type 162-164 

Notation,  rectangular  panels 285,  292 

Poisson's  ratio 168,  169 

Radial  and  circumferential  resilience 136-138 

Side  and  diagonal  belts,  deflection  at  mid  span 199-201 

Stresses  in  column  heads,  formula  for 188-193 

Stresses  in  column  heads,  formula  for 188-193 

Stresses  in  middle  area  of  panel,  formula  for 193-196 

Stresses  in  the  side  belts 181-184 

Size  and  spacing  of  rods 298 

Steel  ratios 78,  83,295,  296,  297 

Test  data  (See  Calculations) 

Theory  of  slabs  unequally  reinforced  in  two  directions 

(a)  Supported  on  columns 277-285 

(b)  Supported  on  beams 285-295 

STAIRS 391-392 

STIRRUPS 117 

SUSPENDED  CEILINGS 396 

TENSION 

Diagonal  and  indirect 87-93,  112,  121-132,  140 

Direct 2,  72 

TESTS 

Method  of  loading 105-107 

For  deflection  (See  deflection) 

Test  of  slabs  (See  slabs) 

Tests  by  Berry  Extensometer  and  Deflectometer 265-268 

WORKING  STRESSES 

For  steel 113 

For  concrete 113 

For  columns. .  318 


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